Head into Milchar and make your way through the tunnels to the Grove of Reflection. They are typically miles away from each other so I will talk about the branch first. It doesn't matter which order you collect the items in.
Ciirta is in the Howling Halls and the branch is in the Grove of Reflection (Milchar). He tells you that you need to find two things to make the Staff the eye of Ciirta and a branch from the Tree of Shades. Head to Knifepoint Hollow which has now been marked on your map and speak to Dyus who is behind a crystal door at the end of the corridor. Speak with haskill and he will tell you that you need to make a new Staff to use in the Font of Madness. When you speak with Sheogorath this time, he will end by transforming into Jyggalag and disappearing.
To unlock this final achievement, you need to complete 3 quests: Symbols of Office, The Roots of Madness and The End of Order. As I have said in my other solutions, make sure to have the difficulty turned right down and to save regularly.