He put a shotgun in his mouth and pulled the trigger. He blew his head off. That's why when the story first broke, it was reported that they thought the reason Kurt Cobain had been missing was that he murdered someone. They didn't realize it was him. He was unrecognizable and was wearing women's clothes. Apparently, they thought he had killed a transvestite. It turns out he was just really high and when he was high he'd often just put on whatever was next to his bed, which a lot of times was a dressing gown of Courtney Love's.
Of course, the other theory is that Courtney Love, that crazy b*tch, blew his head off and got away with it. Because there was a HELL of a lot of shadiness in her story with it changing and changing again and changing again and her never being able to account for her whereabouts. Like, her rockstar boyfriend goes missing and there's this whole manhunt thing going on, and what does she do? She decides to take a vacation to LA in the middle of it--a nationwide manhunt for the father of her baby and rather than look for him or even just hold some vigil at home waiting to find out what happened to him and where he is, she spur of the moment decides it's time to go on a little vacay.