Kreia - Influence Increase - After making your way to Tienn Tubb's shop and speaking to him about changing the Ebon Hawk's transponder codes, you will be approached by his droid with a task. Return to TT-32 with either the droid he requested or it's memory core. With sufficient Intelligence, you will see an option to "negotiate" a discount on Tienn's goods. Successfully close this deal to earn influence for your cunning.
NOTE: You will need to have the Influence Dialog Patch installed before you go on to the next section. Without it, you will be unable to get the airspeeder part off of Lasavvou's corpse. This item is required for your final influence shift with Bao-Dur . If you miss this one, your next (and last) chance to influence Bao-Dur isn't until Dantooine, so you're better off completing your business here.
Atton - Influence Increase - Head over to the Docks and with only Atton in your party, pick a fight with the crew of the Lunar Shadow, the small group of aliens in flophouses. Kill them all for an influence gain. Go around the corner and do the same with Lasavvou, the grounded Ithorian. Finally, go two doors down and murder Lootra. You'll get the third part you need to build your lightsaber as well as an influence increase.
NOTE: Now that Lootra is dead, do not help Aaida clear the way to return to her husband. If you do, Lootra will reappear, Aaida will join him, you'll get Light Side points, and you'll reopen the quest with no way to close it. Your best bet is to let "the bouncers" in the Refugee Quad live, then trick Aaida into trying to make her way to the Docks.
You now have sufficient influence over Atton to make him a Dark Jedi Sentinel, however you must first unlock this necessary dialog option by triggering the cutscene with the two Twi'leks in the Refugee Quad before you can actually do it. Once you've met up with them and learned about Atton's history, you can speak to him again to complete his training.
Now that you've cleaned (flop) house, collect the rest of the pieces that you need to repair the broken airspeeder and head for the Refugee Quad.
Visas - Influence Increase - With only Visas in your party, find Geriel and use your Persuade skills to convince him to commit suicide. After you've taken care of Geriel and convinced Nadaa to give herself over to the Saquesh, you should be able to persuade Hussef to give in to the Exchange. This cunning act will gain you influence with Visas.
NOTE: Exit to the Landing Pad and add Bao-Dur to your party before heading back into the Refugee Quad to repair the broken airspeeder. Upon reaching the Landing Pad, you may trigger a cutscene back at the Ebon Hawk involving the Red Eclipse. Do not go back to the ship at this time. Wait to take care of them until after you have completed your business on Nar Shaddaa's surface.
Also, you may wish to add Atton to your party as well. If you pass through the refugee section on your way to the airspeeder, you will most likely trigger the aforementioned cutscene with the Twi'leks (pass through using the corridor closest to the Serroco section). You will be able to train Atton immediately following that exchange.
Bao-Dur - Influence Decrease - In the Serroco part of the Refugee Quad, you will find the now infamous Broken Airspeeder. While repairing it, you will have a chance to express doubt about Bao-Dur's abilities to fix the airspeeder. This will provide you with the last influence decrease necessary to train Bao-Dur to become a Jedi.
NOTE: You will have to speak to Bao-Dur several times and go through each of the main dialog options available before you can unlock the "training" portion of your conversation.
Finish your business in the Quad, then head back to the Landing Pad. If you've completed everything up till this point, then you will trigger a cutscene involving the Red Eclipse when you enter the Landing Pad module. Get ready for a fight and head back to the ship.
NOTE: When you first enter the Ebon Hawk, you will find yourself in the middle of a fire fight. After you deal with the slavers you will have a few opportunities to gain influence with your party members.
Handmaiden - Influence Increase - If you are at Level 14 or above (and you probably should be at this point), you can initiate the second sparring match for more influence. After this match, she will stop you to question why it is that you choose to go back and face the Jedi Counsel at your trial. Any response that you provide will gain you influence.
NOTE: You now have sufficient influence to train Handmaiden, however just as you had to trigger the conversation with the Twi'leks before you train Atton, you must complete the last sparring match with Handmaiden before she can become a Dark Jedi Guardian.
Kreia - Influence Increase - Speak to Kreia and indicate that you wish to continue the Lesson of Listening. When you are finished with you gain more influence with her as well as some other rewards for completing the lesson.
NOTE: Now that you have completed the Lesson of Listening, you can speak to Atton about his mental Pazaak habits to gain an increase to your Will saving throws.
You're now finished with all the influence opportunities that are available to you at this time. Once you leave the Ebon Hawk, you will trigger the rendezvous with Visquis in the Jekk'Jekk Tarr.
NOTE: By the time that you return from your trip to Goto's Yacht, you should have 3 new party members: G0-T0, HK-47, and Hanharr. Considering HK-47 a party member is a stretch because technically, you will only have all of the parts that you need to finish repairing him and not the completed droid himself, but let's not split hairs. All three of your new party members lean heavily toward the Dark Side, so you should be able to max your influence with them pretty easily.
Wrap up any loose ends on Nar Shaddaa and head back to the Ebon Hawk to finish your repairs on HK-47.