What episode did Goku kill buu?

Goku didn't directly kill Majin Buu. Here's how it went down:

* Goku and Vegeta fused into Vegito and fought Buu, but Vegito was eventually absorbed.

* Inside Buu's body, Goku and Vegeta separated and used the Spirit Bomb to destroy Buu from the inside.

* The Spirit Bomb was not technically "killing" but more like erasing Buu's existence.

So, the episode where Buu was destroyed is Dragon Ball Z Episode 288 (English Dub) or Dragon Ball Z Episode 287 (Japanese).