The Duel Ring
In the Upper City's Cantina, you will find a dueling pit where you can earn a few extra credits by taking on challengers in non-lethal duels. The challenges start off fairly easy, but steadily ramp up in difficulty. If you want the credits for the earliest matches, you can fairly easily make it through a few rounds before heading into the Lower City, but you'll probably want to save the tougher enemies until you return from the Under City with Bastila in your party; her cure skill can save you a fortune in medpacs, and Brejik's belt and armband, which you should've found on Brejik's body after the swoop race, can save you a lot of melee damage.
Deadeye Duncan: Deadeye is intended to be the infant-level difficulty contender, so just cream him a couple of times with any weapon and he'll keel over. You get 100 credits from the fight.
Gerlon Two-Fingers: Another powderpuff opponent. Gerlon'll use blaster pistols from a distance, then switch over to a melee weapon if you attempt to go toe-to-toe. Pick up 200 credits for the win.
Ice: She'll use a blaster pistol from a distance, then switches over to a vibroblade if you close in on her. Your reward is 300 credits.
Marl: Marl's going to be the first real challenge, especially for scoundrels who may not have enough vitality to live through the contest. He uses a dual-bladed vibroblade with master critical strike, gets two attacks per turn, and has +10 to attack and +4 to damage. Needless to say, you're gonna be taking a beating. His weapon can sometimes stun you, and if this happens, you can start reloading right then, because you're not likely to survive the two rounds or so it takes for the stun effect to wear off. Weaker characters might just not be able to get past him (though you might try some creative grenade trickery before he closes in), but soldiers and scouts should be able to take him down. You'll get 400 credits for the win.
Twitch: Twitch initially comes at you with dual blasters, with mainhand stats of +12 attack and +2 damage, and offhand stats of +10 attack and +2 damage. If you close in on him, though, you can force him to whip out a vibroblade, with much less impressive stats of +10 to attack and +1 to damage, with a single attack per round; this is obviously the way to go if you are comfortable with melee combat yourself. Ranged attackers might want to use grenades to soften him up or stun him, since the damage he deals with the blaster pistols can add up quite quickly. You'll get 500 credits for becoming the Duel Ring champion.
Bendak Starkiller: Bendak will only agree to fight you after you've defeated the previous five candidates that Ajuur sets up for you; all things considered, this is a good thing, since he's much tougher than any of the softies that play the simulated duels. Bendak's only interested in the real thing: a deathmatch, and we're not talking Quake 3.
After speaking to both Bendak and Ajuur, leave the cantina, and then re-enter it immediately afterward. Talk to Ajuur to start the match. You'll take an immediate Dark side hit for agreeing to participate in this kind of savagery.
Bendak starts out by chucking plasma grenades at you, so run to either side to avoid their effects. After the third grenade, he'll whip out his trusty blaster pistol and use his +18 attack with master rapid shot and master power blast to pelt you with attacks both powerful and numerous; if you're going to exchange blaster fire with him, now would be a good time to activate an energy shield. Melee combat against Bendak is only slightly less dangerous, since his +15 to attack and +3 damage is coupled with master power attack. His 21 defense will be tough for your character to hit in the best of circumstances unless you've levelled quite a bit.
One opportunity to dish out maximum damage may be to try, as with Twitch, to pelt him into submission with grenades. If you can keep an energy shield up long enough, you should be able to shower him with grenades, which should severely weaken him, if not kill him, assuming you keep it up long enough.
After Bendak falls, talk to Ajuur one last time to get 700 credits that were wagered, as well as Bendak's blaster. If Carth is in your party, you might want to warp back to your apartment and upgrade Bendak's blaster to take the place of whatever you're using in his off-hand.