Rewatch the film. They don't mention the Eugenics Wars but they do mention him trying to lead the world and towards the end Spock mentions that if he gets the seventy-two other Augments, that he would again start trying to cleanse those that don't meet their standards.
Well, "backstory" may not be the appropriate term for the distinction Lapis Exilis
was originally contemplating.
It's possibly more that what we see feels like a very incomplete chunk of Khan's story.
True but I don't think that any more was necessary because what we see of Khan in the movie itself is much more terrifying than someone making vague remarks about the Eugenics Wars like in "Space Seed".
Khan doesn't even flinch when he's accused of wanting to commit genocide on those he considers inferior, unlike Khan in "Space Seed" who tried to justify his past actions.
We see what destruction he alone can cause, first on Qo'nos, then onboard the Vengeance. Just imagine what 72 of those people could do.
I've been a life-long Trekkie and I didn't mind Khan looking or being different from the one we saw in TOS. To me, this is all a different take on themes and characters from Trek lore.