The Imperial City Arena is Cyrodiil's theatre of blood and pain, a place where the best warriors cross their blades fighting to death. You can sign up by talking to Owyn, standing somewhere in Arena Bloodworks. At first, you'll be mocked and laughed on, but with each victorious fight your prestige will rise and so will your rewards.
The Arena matches are none too easy, the opponents are always a few levels above your own. You can fight any time between 9 A.M. to 9 P.M.
When fighting, you must wear an Arena Raiment. You'll get it from Owyn when registering for the first time. You'll have to choose whether you want to use Light or Heavy Raiment. When choosing, just keep in mind in which armor your character actually feels better. The arena is not a place to practice new skills, there's no time to hold back.
Because you can't use your own armor, try to make up for it by wearing the best helmet, shield, weaponry and jewelry you can get.
Before each match, you have to tell Owyn that you're ready. After the fight you go back to Arena Bloodworks, freshen up in the fountain, and talk to the blademaster once again, to get your reward and, each three matches, an advancement in rank.
HINT: Due to a game bug, it's quite easy to win any match against any enemy that doesn't have a ranged attack. Just hop on the shelf shown on the screenshot below (first to the lower, then to the upper level). When you're standing there, no melee attacks can reach you, and you can easily use arrows or magic to kill your opponents.
Road to glory
When you are named Champion, the only challenge left is to fight Agronak gro-Malog, the Grand Champion. Ask Ysabel Andronicus about that, and challenge Agronak personally. He'll accept it. This time the rules are a little bit different: you can use any armor you like, and after you win you get to take Agronak's equipment.
Ysabel will tell you to choose your nickname (heh) and prepare for the worst. Enter the arena, warrior. What happens now depends on whether you have performed a quest called 'Origin of the Gray Prince' (given by Agronak himself). If you have, the half-orc will just stand there and beg you to kill him, so ashamed of his heritage. If you haven't, there's a difficult battle before you. When you win, take his armor and take it to Ysabel to get your own super-raiment.
As the Grand Champion, virtually unmatched throughout Cyrodiil, you won't receive any challenges. Ysabel will tell you, though, that you can both benefit from your abilities by organizing special shows, where you will be fighting some wild beasts. You can participate in such event once a week - and get some decent money from it.
The second perk is that you actually get to have an adoring fan. When leaving the arena, you'll be accosted by a funny Wood Elf, who offers his service as your everyday bootlicker. Should you agree, he'll follow you around, not doing anything particular. There's nothing special to him - treat him more like an Easter Egg.