The Elder Scrolls IV: Oblivion/Imperial City Quests

Buy a house in the Imperial City
See here.

Order of the Virtuous Blood
After completing the main story, a lady will visit you, and tell you that her husband's friend wants to speak to you. You must carry out the tasks he wants you to do.

Origin of the Gray Prince
Go to the Arena, and defeat the Champion. Talk to the Gray Prince, and he will ask you to do him a favor. Accept, and he will give you a key, which begins the quest.

Unfriendly Competition
Talk to Jensime in the Imperial City (she runs Jensime's "Good As New" Merchandise in the Market District). She will make a passing comment about a guy named Thoronir. Ask her about him, and she will explain that Thoronir's prices are absolutely insane, and that she's certain that his merchandise is stolen. Agree to help her, and wait outside The Copious Coinpurse (Thoronir's shop) until he leaves. Follow him around for a while and you'll witness him meeting with a suspicious Nord named Agarnir.

After the meeting, start following Agarnir, and you'll eventually end up at his house. Start waiting until Agarnir leaves,(which is around 5pm) then sneak into his house (making sure the guards don't see you). Go down and search the basement until you find what you're looking for.

You should go into his basement and you will find loads of bonemeal and skulls, you will also find a book listing dead people, pick this up and take it back to Thoronir.

Thoronir will be disgusted and will tell you he said he was going somewhere tonight. Follow your objective indicator and go into the tomb, he will say he dug a grave just for you and will try and kill you.

When you beat him, take the Shovel from the grave. Give it to Thoronir then go to Jensime for your reward.