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.
Prior to the beginning of the film, we are to assume that, from memory: he was awakened from the cryo-sleep by Admiral Evil, is betrayed by him, and is seeking revenge. That is where Khan's story overlaps with the movie.
It's an awkward way to introduce the character. No doubt that is partly because they didn't want to overlap with the events of Space Seed. But that is why it feels reliant on knowing the previously established story.
In TWoK, the main relevant piece of information is established in the film itself: Khan having been marooned by Kirk and escaping.
It's a necessary by-product of the film using both Khan and Marcus/Section 31, and it's not a huge deal, but I think it does make an impact. In essence, we are getting exposition, not only about Khan's backstory prior to being frozen, but also *after* being un-frozen, so there is a lot of "tell."