Dragonborn by lore are mostly found in Tymanther or far eastern Faerun, their largest population is on Laerakond. They are actually very rare in the Sword Coast region, you are lucky is see more than one or two in a life time with maybe the exception of Waterdeep.
They tend to, socially, be very honorable people. "Dragonborn had a well-known dedication to honor, but how deeply rooted this trait was remains debatable. However, all dragonborn tended to view all living things, even hated enemies, as deserving of courtesy and respect. To dragonborn, honor was more than a word and was often considered more important than life itself. Cowardice was not simply undesirable among dragonborn, it was considered outright repulsive."
As for classes:
The race's natural abilities were well suited for the path of a warlock, though few dragonborn were willing to actually take up the career since it forced them to deal with powers beyond their control.
Dragonborn arcane spellcasters were, however, far more often wizards or sorcerers, attracted by the often secretive and eccentric ways that both paths involved.
Dragonborn who did embrace religion, however, were well-suited to the path of a paladin. Making the creed and commandments of a god a code by which they could live was in many ways attractive to dragonborn, so long as they were able to get past their natural aversion to beholden to a superior authority they had no control of. If they did join the faithful, dragonborn could become quite devoted in their chosen church, craving the direction and purpose it gave them. Likewise, paladins, unlike clerics, represented an almost martial career path and one that blended well with the militaristic culture of the race.
Found an article about best Dragonborn classes in 5 and they suggest: Sorcerer, Bard, Barbarian, Fighter and Paladin (this is number 5 to number 1)
