To cleanse the Essence of Vanity you require the Horned Wreath. Cleansing the Essence of Vanity will net you the Tranquility sniper rifle.
1. Make your way to the Moon.
2. Spawn at Sanctuary
3. Pop your sparrow.
4. Head out the right exit (not the one through the cave)
5. Head to the Anchor of Light and then go through to the Hellmouth.
6. Enter the Hellmouth.
7. As you go down the inside platforms there will be a hole in the wall that lets you go deeper.
8. Continue down the stairs.
9. Keep going down.
10. You should enter a series of caves, a desolate room and then another series of caves.
11. Keep going and you should enter the Circle of Bones
12. Continue and you should enter The World's Grave
13. Go right through the door.
14. Keep going straight and you will be in the Chamber of Night
15. Move straight and you will eventually see a location marker pop up. An Ogre will likely be guarding the wreath. Dispose of him and click on the Wreath will be designated a "Weapon Part".
Here is a great video by Esoterickk that takes you through this long, but relatively straightforward run.
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