As previously discussed, Khan is the product of a eugenics project on Earth in the late 20th century. It produced not only Khan Noonien Singh, but a number of other genetically enhanced people as well. They set themselves up as dictators over a great deal of the planet, and are only stopped by the Eugenics Wars of the 1990's. Khan and a bunch of his comrades manage to escape into space aboard the Botany Bay, in suspended animation until the Enterprise stumbles upon them and thaws them out.
** So that's my look at Kirk vs. Khan. I'll have one more post on The Wrath of Khan, discussing the themes of the film.