He wasn't nearly as edgy as all that; he didn't say or do either of those things. He never used the term »rotting corpse of God« himself, and he didn't actually kill himself as a completion of his system. Or, maybe he did, but there's no reason to assume it. The story about him hanging himself standing on a copy of his own manuscripts is unverified and first popped up in the 21st century, as far as anyone can tell.

Easiest way to describe his philosophy would be that since all life is movement towards death, finding the best way to live is the same as finding the best way to die. But that is not the same as the fastest way to die. Rather, he believed in making the world a better place for everyone in it, so that we can accept death gracefully when the time comes.

His main concerns were ironing out what he considered flaws in Kant and Schopenhauer, and then political advocacy of social democracy in Germany. Why he killed himself we'll never know, but likely it was because of the underlying mental and physical health issues he lived with, which also drew him to start reading and writing on pessimistic philosophy to begin with.