1. After completing Hear me Roar, go to the castle of King's Landing and talk to Jacelyn Bywater. He will tell you about the melee which is held for King Joffrey's entertainment. Sign up for the melee.
2. Go to the melee grounds, talk with the steward.
3. Press TAB, then join the fight. Withdrawing from the melee will fail the quest.
4. You'll have to defeat six opponents in 1v1 duels, each next one more difficult than the previous one. The last opponent is Sir Sandor "The Hound" Clegane himself, a very tough opponent who can knock out unprepared character in one hit. Losing any fight will fail the quest.
5. After defeating Clegane, you'll join the celebration in the castle. You may celebrate your victory and get wasted, or avoid drinking. The latter option will increase relations with King's Landing and Tywin Lannister, while getting drunk won't give any rewards nor incur any penalties.
The rewards are 10 000 coins and 70 Renown.