How can I kill my goat?

You're out of luck, you can't die in Goat Simulator.
You're an immortal soul - once human, maybe - doomed to be alone, caught in purgatory.

(This took a rather bleak turn.)

You can't die in Goat Simulator. The goat is invincible and will always bounce back up after rag-dolling.

It's possible in some versions of the game* to clip out of the world, thus despawning your goat and hanging the game, using Evil Goat's "attractor" ability whilst ragdoll in order to spin-fly very, very high and travel through the top of the skybox (skydome?). Turning the ability on and off with the right timing appears to grant you some level of control over the process, but I've only managed to do this successfully twice.

Notably, you cannot achieve this by clipping out of the side or the bottom of the world, since this results in you teleporting back to the spawn point.

The goat dying is not implemented into the game, so it is (as far as I can tell) impossible to recover from such an event.

^{*: i.e. I've tested this on the iPad version only}