Looking over the Player's Handbook for 5e I have decided to make a little table and overview of which backgrounds mix well with the main classes and pair them up for getting the most out of your chosen class should you decide to make your own custom character, and as a way to help new players get accustomed to a BG game set in 5e rules.

WIS=Wisdom STR=Stength DEX=Dexterity INT=Intelligence CHA=Charisma CON=Consitution



Remember that this list is just a suggestion and a means to help people think about how to make the most of the game. It is not a guide, mix and match as you see fit.

Class: Barbarian
Saving Throws: Strength and Consitution
Skills: Choose two from Animal Handling, Athletics, Intimidation, Nature, Perception, and Survival
Suggested Background: Outlander
Skills proficiency gained from Background: Athletics (STR), Survival (WIS)

Class: Bard
Saving Throws: Dexterity and Charisma
Skills: Choose any three
Suggested Background: Entertainer
Skills proficiency gained from Background: Acrobatics (DEX), Performance (CHA)

Class: Cleric
Saving Throws: Wisdom and Charisma
Skills: Choose two from History, Insight, Medicine, Persuasion, and Religion
Suggested Background: Acolyte
Skills proficiency gained from Background: Insight (WIS), Religion (INT)

Class: Druid
Saving Throws: Intelligence and Wisdom
Skills: Choose two from Arcana, Animal Handling, Insight, Medicine, Nature, Perception, Religion, and Survival
Suggested Background: Hermit
Skills proficiency gained from Background: Medicine (WIS), Religion (INT)

Class: Fighter
Saving Throws: Strength and Constitution
Skills: Choose two skills from Acrobatics, Animal Handling, Athletics, History, Insight, Intimidation, Perception, and Survival
Suggested Background: Soldier
Skills proficiency gained from Background: Athletics (STR), Intimidation (CHA)

Class: Monk
Saving Throws: Strength and Dexterity
Skills: Choose two from Acrobatics, Athletics, History, Insight, Religion, and Stealth
Suggested Background: Hermit
Skills proficiency gained from Background: Medicine (WIS), Religion (INT)

Class: Paladin
Saving Throws: Wisdom and Charisma
Skills: Choose two from Athletics, Insight, Intimidation, Medicine, Persuasion, and Religion
Suggested Background: Noble
Skills proficiency gained from Background: History (INT), Persuasion (CHA)

Class: Ranger
Saving Throws: Strength and Dexterity
Skills: Choose three from Animal Handling, Athletics, Insight, Investigation, Nature, Perception, Stealth, and Survival
Suggested Background: Folk Hero
Skills proficiency gained from Background: Animal handling (WIS), Survival (WIS)

Class: Rogue
Saving Throws: Dexterity and Intelligence
Skills: Choose four from Acrobatics, Athletics, Deception, Insight, Intimidation, Investigation, Perception, Performance, Persuasion, Sleight of Hand, and Stealth
Suggested Background: Criminal or Urchin
Skills proficiency gained from Background: Deception (CHA), Stealth (DEX)/Sleight of Hand (DEX), Stealth (DEX)

Class: Sorcerer
Saving Throws: Saving Throws: Constitution, Charisma
Skills: Choose two from Arcana, Deception, Insight, Intimidation, Persuasion, and Religion
Suggested Background: Guild Artisan
Skills proficiency gained from Background: Insight (WIS), Persuasion (CHA)

Class: Warlock
Saving Throws: Saving Throws: Wisdom, Charisma
Skills: Choose two skills from Arcana, Deception, History, Intimidation, Investigation, Nature, and Religion
Suggested Background: Charlatan
Skills proficiency gained from Background: Deception (CHA), Sleight of Hand (DEX)

Class: Wizard
Saving Throws: Saving Throws: Intelligence, Wisdom
Skills: Choose two from Arcana, History, Insight, Investigation, Medicine, and Religion
Suggested Background: Sage
Skills proficiency gained from Background: Arcana (INT), History (INT)

Also available as a Background is a Sailor. With Skill proficiencies in Athletics (STR), Perception (WIS), just to cover all my bases.

Also be aware that most custom characters are native of the City of Baldur's Gate (Baldurans) unless they are a race native to the Underdark (drow/half-drow) or from another plane, i.e. Githyanki from the Astral Sea.

A follow up to the above is a clarification on what the skills are and how they can help you in the game.
Acrobatics
Modifier Type: Dexterity

Situations of Use: Your Acrobatics skill encompasses your ability to stay on your feet in a tricky situation. The following are examples of when you might be asked to make an Acrobatics check:

Running across a sheet of ice
Balancing on a tightrope
Staying upright on a rocking ship’s deck
Performing acrobatic stunts, including dives, rolls, somersaults, and flips
Animal Handling
Modifier Type: Wisdom

Situations of Use: According to the PHB, Animal Handling only relates to making checks with domesticated animals. These checks might include:

Calming down your mount if they have been frightened
Determining if your mount could make a jump
Arcana
Modifier Type: Intelligence

Situations of Use: Your Arcana skill reflects your knowledge in the realm of spells, magic items, eldritch symbols, magical traditions, the planes of existence, and the inhabitants of those planes.

Athletics
Modifier Type: Strength

Situations of Use: Athletics encompasses your ability to succeed in difficult situations you encounter while climbing, jumping, or swimming. Examples include the following activities:

Attempting to climb a cliff
Clinging to a surface while something is trying to knock you off
Jumping an unusually long distance or pulling off a stunt midjump
Struggling to swim or stay afloat in treacherous currents or something trying to pull you underwater or otherwise interfere with your swimming
Deception
Modifier Type: Charisma

Situations of Use: Your Deception skill encompasses your ability to convincingly hide the truth, either verbally or through your actions. This deception can encompass the following:

Trying to fast-talk a guard or con a merchant
Passing yourself off in a disguise
History
Modifier Type: Intelligence

Situations of Use: Your History skill measures knowledge about historical events, legendary people, ancient kingdoms, past disputes, recent wars, and lost civilizations.

Insight
Modifier Type: Wisdom

Situations of Use: Your Insight skill encompasses your ability to read the true intentions of a creature, such as when searching out a lie or predicting someone’s next move.

Intimidation
Modifier Type: Charisma

Situations of Use: When you attempt to influence someone through overt threats, hostile actions, and physical violence, the DM might ask you to make an Intimidation check.

Investigation
Modifier Type: Intelligence

Situations of Use: When you look around for clues and make deductions based on those clues, you make an Investigation check. Examples would include:

Deducing the location of a hidden object
Discerning from the appearance of a wound what kind of weapon dealt it
Determining the weakest point in a tunnel that could cause it to collapse
Medicine
Modifier Type: Wisdom

Situations of use: Your Medicine skill encompasses your ability to stabilize a dying companion or diagnose an illness.

Nature
Modifier Type: Intelligence

Situations of use: Your Nature skill measures your knowledge about terrain, plants and animals, the weather, and natural cycles.

Perception
Modifier Type: Wisdom

Situations of use: Your Perception skill measures your ability to spot, hear, or otherwise detect the presence of something. It measures your general awareness of your surroundings and the keenness of your senses. Examples of this would include:

Trying to hear a conversation through a closed door, eavesdropping under an open window, or hearing monsters moving stealthily in the forest
Spotting things that are obscured or easy to miss, such as whether there are orcs lying in ambush on a road, thugs hiding in the shadows of an alley, or candlelight under a closed secret door
Performance
Modifier Type: Charisma

Situations of use: Your Performance skill determines how well you can delight an audience with music, dance, acting, storytelling, or some other form of entertainment.

Persuasion
Modifier Type: Charisma

Situations of use: When you attempt to influence someone or a group of people with tact, social graces, or good nature, the DM might ask you to make a Persuasion check. Persuasion typically comes up over Deception when you are acting in good faith, fostering friendships, making cordial requests, or exhibiting proper etiquette.

Religion
Modifier Type: Intelligence

Situations of use: Your Religion skill measures your knowledge about deities, rites and prayers, religious hierarchies, holy symbols, and the practices of secret cults.

Sleight of Hand
Modifier Type: Dexterity

Situations of use: Sleight of Hand determines your ability to plant something on someone else without them knowing or to conceal an object on your person. The DM might also call for a Sleight of Hand check to determine whether you can lift a coin purse off another person or slip something out of another person’s pocket.

Stealth
Modifier Type: Dexterity

Situations of use: Your Stealth skill measures your ability to conceal yourself from enemies, slink past guards, slip away without being noticed, or sneak up on someone without being seen or heard.

Survival
Modifier Type: Wisdom

Situations of use: Your Survival skill encompasses your ability to survive in the treacherous wilderness. This could include:

Following tracks of game or a monster that you are hunting
Guiding your group through frozen wastelands
Avoiding quicksand and other natural hazards

Remember to pick a background that will compliment your playstyle and provide you with a useful skill in your adventure.

Good overview for folks - esp those less familiar with D&D and 5e. Nice job, Ascendent!

An explanation and overiew of the backgrounds in the Player's Handbook in DND 5E.

Acolyte: You are a part of, and possibly even raised by, a church or cult. You gain proficiency in Insight, naturally, Religion, and your background feature lets you ring up churches of your faith for free help for your party and freeload on their couch. It also lets you pick a single specific temple, somewhere in the world, where you can expect free assistance.
Charlatan: A con-artist background, complete with proficiency in forgery and disguise kits. Naturally, it comes with proficiency in Deception and Sleight of Hand, a list of possible cons you're good at running, and a feature that lets you switch to a fake identity when the heat's on, as well as perfectly copy handwriting and statuary so long as you have a sample to study.
Criminal: Just a regular ol' crook, though you can pick which kind from a list that includes blackmail, smuggling, burglary, or just plain hired murder. Proficiency in Deception and Stealth (what a surprise), with a background feature that gives you a reliable, trustworthy contact in the criminal underworld, and the ability to stay in touch with him wherever you go. (Honor among thieves and all that.)
Spy: A "variant" criminal, who gets the same skillset, but uses it for espionage. Might be a loyal servant of your government or a free agent, and your "contact" is now your "handler."
Entertainer: ...It's not just the bard background, okay? Covers a wide variety of different kinds of entertainer, from musician to storyteller to actor to fire-eater. Gains proficiency in Acrobatic and Performance, and has a fun background feature that lets them perform somewhere for free room and board, as well as improving their reputation in places they've performed before.
Gladiator: A "variant" entertainer, who engages in a flashy kind of "performance" combat. They use their background feature in fight pits and arenas and get free medical care from battling there. Also, luchador or no guts.
Folk Hero: One of the more "freeform" backgrounds. A folk hero is a lowly peasant kid who did something cool that sent them off on a life of adventure. You can pick what from a list or make up your own. You're proficient in Animal Handling and Survival, thanks to your blue collar roots, and the common people and peasantry love you and will do anything short of dying to protect you. You also gain proficiency in vehicles (land), so have fun in high-speed wagon chases.
Guild Artisan: Not really a wild artiste type so much as a union man (or woman) through and through. You are proficient in not only the tools of your trade, but in Insight and Persuasion, because art is good and all but you gotta be good at selling it too. Your background feature gives you a huge list of benefits, and the ability to cozy up to bigger boys in the guild with donations of money and magic items: so long as you're in the union, the union's got your back. However, you also have to keep paying your union dues, or the union's gonna cut you off, so better hope you aren't stuck on one of those lame wilderness adventures where half the "social" background features are useless, because then you'll have to keep track of backpay.
Guild Merchant: A variant, representing someone who buys and sells rather than manufactures and sells. You can swap out your tool proficiency for proficiency with navigator's tools, a vehicle, or an extra language, and can swap out your starting tool for a mule and cart.
Hermit: Somebody who spent time in seclusion from the secular world, whether as a monk looking for spiritual enlightenment or a reclusive inventor or artist. You are proficient in Medicine and Religion (presumably from all that contemplation), and your Background feature is one of the weirdest in the game: some kind of "discovery." Exactly what this is supposed to do is maddeningly inexact, and it doesn't have a list of actual benefits like most of the other backgrounds, so much that the book outright admits that you should sit down with your DM and try to figure out what it actually does.
Noble: The upper-crust background, covering all manner of nobles from the militaristic to the foppish. Proficiency in History and Persuasion from your classical education, and your background feature not only gives you some basic social benefits from being privileged, and you can get audiences with local nobles.
Knight: Even moar military nobles. Instead of the noble's many social benefits, you gain three henchmen who can help you do things. They don't fight, and they'll bolt if you treat them like shit (meaning this is theoretically one of the few background features that can be permanently lost, although if that happens your DM is either a dick or you deserve it). But beyond that, they can do a lot of other mundane tasks, and can be given your signet ring and such to grease the wheels for the party.
Outlander: The ranger to the Entertainer's bard, the Outlander is a backwoods country bumpkin who's used to hard living in the wilderness. Offers proficiency in Athletics and Survival, and has a background feature that, unlike the social benefits of others, improves your ability to navigate and lets you feed yourself and up to five people without skill checks required, so long as it's not an area of wasteland where nature's bounty is too scarce to allow that.
Sage: The "went to college" background, meaning you can pick your major. You're proficient in Arcana and History, and your background feature gives you the ability to recall where, exactly, you can go to try to unearth information you might need, though it notes that this power can sometimes just tell you "I 'unno, lol." Most DMs will, hopefully, not screw you out of your background's big feature unless you try to ask for the secrets of the cosmos or something.
Sailor: Just what it sounds like. Offers proficiency in navigator's tools and water vehicles along with Perception and Athletics. Your background feature lets you easily secure passage for your party during sea travel via your waterside connections, though doing so means that you'll have to help out along the way.
Pirate: The obvious sailor variant. Instead of the sailor's background feature, you've got a reputation, deserved or not, as a mean son-of-a-bitch, meaning that you can get away with some minor criminal offenses, like breaking down doors or skipping out on your tavern bill, because people are too scared of you to confront you or to call the cops.
Soldier: Hoo-hah. Another beautiful day outside of this man's army. You're proficient in Athletics and Intimidation, and you still have your old military rank, meaning you can throw some orders around and lower-ranking people will obey, you can temporarily requisition supplies like horses and carts, and you can usually get access to friendly military fortresses and checkpoints.
Urchin: It's a hard knock life. You grew up on the mean streets, and to get by you got proficient with Stealth, Sleight of Hand, and thieves' tools. You also have a cute pet mouse and a memento of your poor dead, dead parents, because dawww. Your background feature lets you easily navigate urban environments, moving through them twice as fast as normal out of combat.

The image at the top is nice, but there is one problem I have with it:
Strengh, dex and so on are attributes ( which are called abilities in 5E to add confusion), not skills.
For 5E the image should replace the term "skills" with "abilities" or change it to "athletics lets you crush tomatoes, acrobatics lets you dodge tomatoes, . . ." and so on.