Spock does the research off-screen. He reports to Kirk the results of this research. The general audience member will go, "Oh, that is who Khan is. Okay, now I get why he is such a big threat."
Because Khan pulling a John Rambo on Starfleet headquarters and a platoon of heavily armed Klingon troopers wasn't good enough
Anyone who wants to know who Khan ACTUALLY is can do what my sister/friends/wife/cousins/boss/daughter did and watch Space Seed on Netflix. Anyone who wants to know how the Enterprise ACTUALLY beat Khan last time around can watch Wrath of Khan. Anyone who wants to know why they need to be worried about Khan simply needs to not miss the first half of the movie.