Hmm, your character is still killing someone who has done them no harm and shows no sign of being a threat because of some general attitude to goblins as a whole, which seems racist to me. Or speciesist, as it may be.
Speciesist isn't a problem
It's okay to shoot zombies. In fact it's a problem if you see a zombie and don't shoot it -- if you just let it shuffle on its merry way it will probably eat someone's brain. Shooting members of homo necromancis is a good thing to do!
Same goes for goblins. They're born with black hearts. Their culture doesn't make them that way - their souls were claimed by their dark gods before birth. They are all lawful evil because their gods are all lawful evil:
https://www.dndbeyond.com/monsters/16907-goblinOf course neither zombies nor goblins exist in the real world. In real life you shouldn't shoot or stab . . . well . . .anyone! Anything! Please put the swords down.
In the real world, on earth, racism is a problem because it relies on the scientifically false belief that the human 'race' is divided into species or subspecies. But, again, that's not true - we're all homo sapiens. Racial prejudice makes judgments on the value of the different human 'races'. And that's a problem because it leads to real world harmful practices. Unfortunately, language has slipped and people no longer make a distinction between racism and racial discrimination. But the prescriptivist in me hates that decriptivist change in the language . . .
DnD uses race in it's original meaning - as ye olde word for species. DnD is a fantasy world where different intelligent species do in fact exist and some are better than others and it's okay to commit acts of violence against evil species. Fantasy =/ reality and it's a problem to confuse them. Which is why it's good to retire the world race - so people won't conflate fantasy with real life
Now can you subvert these rules? Of course! I thought the zombie romance movie was kinda sweet. Can you make a goblin / demon / skeleton adventurer? Sure! Why not. The rules aren't written and stone and your DM can do anything. But I don't want to see the entire world altered where it becomes bad to stab zombies because that zombie could be someone's boyfriend. Neither do I want to see a world where it's bad to wipe out a camp of goblins because we know that goblins "
gather in large numbers to torment other creatures" In DnD if you kill a goblin, a zombie or a demon you are probably saving another innocent life.
So TL;DR - in DnD you can't be 'racist' because believing that different intelligent species exist is not false. Neither is believing that some races are better than others a problem, it's just the 'reality' of the fantasy setting. And people who kill evil species aren't ice holes, they're adventurers doing their bit to make the world a better place.
Now that may be about to change in 6e - if it does I'll probably seek out a new ruleset to obsess over.
