The arena is one of the easiest ways to make money and you can use it from the start of the game. All you have to do is fight enemies in combat. You will eventually gain ranks which comes with a bigger prize for killing the enemies. To start, go into the bloodworks and talk to the Blademaster, get the armour, put it on and then ask him if you can fight. You can do anything in the arena apart from looting peoples bodies after you win. So spells, summoning and sneaking are all allowed.
Your first battle will be against a swordswoman, in all matches, when they are far away from you, use your bow or spells to attack the enemy before they get near, this weakens them, this fight is fairly easy, block, then when they hit you, take a few swings at them, repeat this until they are dead, now go to the blademaster and collect your reward of 50 gold. Don't forget to drink form the basin once you finish to restore your health, magicka and fatigue
The second battle will be similar to the first, use the same strategy and you should win and don't forget to use restoration spells if you get into trouble as this enemy hits a bit harder. Collect your 50 gold when you win.
Your third match is an advancement match against a bowman, the best idea is to get in close and kill him with your sword, you should block while advancing to stop all of his arrows and maybe have a bit of weight left as the arrows may just add up your encumbrance. After you block an arrow, you may want to fire a spell or an arrow at him. After you win the match, you will advance to brawler rank.
Origin of the Gray Prince Quest
For the quest information click here