Old English huntian "chase game," related to hentan "to seize," from Proto-Germanic *huntojan (cf. Gothic hinþan "to seize, capture," Old High German hunda "booty"), from PIE *kend-.
General sense of "search diligently" (for anything) is first recorded c.1200. Related: Hunted; hunting. Happy hunting-grounds "Native American afterlife paradise" is from "Last of the Mohicans" (1826).