"Kill" has quite a few meanings in colloquial English. As Ruthi has said, it can be in reply to a joke. It can both be positive and negative. "You kill me" can be said in response to a good joke, or to a bad one - "you're killing me man..." It comes out a bit sarcastic, a bit sardonic. It's also seen to be used in forms like "killer" - "What a killer question", meaning "what a great question". "That's killer!" - "That's awesome!"

I remember saying things like: he absolutely killed me! (he was so funny I nearly died laughing). And as it is often people's WORDS that are funny, I suppose you could say, "His/your words killed me." I don't see why not. Often I start the sentence with: "It absolutely killed me when.... (he fell off his bike into a puddle/etc.)

Not exactly like that. But we can, in slang say "Oh, you kill me" after someone says something funny. It is quite common.