HK-47 - Influence Increase - After landing in the Iziz Spaceport, head to the entrace to the Merchant Quarter, where you will be stopped by a guard. Use the Force Persuade options to get by. When HK-47 comments, reply with "It's inside a city, HK, otherwise...". Afterwards, ask him "Why did Revan not consider the Republic a threat?". Follow along until you have an opportunity to ask him "If I need to kill a Jedi, how would I do it?". This branch of dialog will unlock HK-47's considerable rewards.
Visas - Influence Increase - Add Visas to your party and then head into the Merchant Quarter. Find the aliens debating politics by the entrance to Western Square. When they ask your opinion and Visas interjects, reply with "Vaklu sounds capable and has ambition. Drive and experience is a powerful combination.", "The Republic has taken advantage of Onderon. A little revolution, now and again, can be a good thing.", or "I'll lend support to whomever serves my interests better."
G0-T0 - Influence Increase - Head into the Western Square and find the droid merchant, 1B-8D. Try to hack into his systems for a discount and wait for G0-T0 to offer his assistance. Reply with "Your assistance would be appreciated" to gain influence with him.
NOTE: You may want to remove G0-T0 from your party before heading back to the Merchant Quarter. If you do not, he will threaten the bounty hunters that try to ambush you, thereby robbing you of some XP and some valuable loot. Most importantly, you will need the group leader's Starport Pass so that you can alter it with Kiph (and in turn trade it with Sakarie). After you've made your way to the Merchant Quarter, add him back to your group.
Once you're back in the Merchant Quarter, go towards Ponlar and his growing group of followers. Things have come to a boil, so use your Persuade skill to rally the crowd around him. G0-T0 will chide you about instability, but reply with "Vaklu ultimately would bring greater stability to Onderon. We need to demonstrate we're willing to work with him." to bring him back to his senses.
Finally, head to the Iziz Cantina. Take the Starport Visa that you found on the bounty hunter and pay Kiph to alter it for you. Turn around and offer to trade it to Sakarie for whatever you'd like. During the transaction, G0-T0 will once again offer his opinion on the matter. Respond to his comment with "It's the only way to make a profit."
NOTE: Believe it or not, this is all the influence that you need to open up G0-T0's backstory. Congratulations; you've done all you can with him.
Visas - Influence Increase - After finishing your business with Dhagon Ghent and sending your message to the Palace, go back to the Cantina with Visas in your party. Once Tobin crashes your reunion with Kavar, Visas will be ready for some good old fashioned brawling. Tell her to "cleanse away" for the final influence boost that you'll need to unlock her rewards.
NOTE: Due to some unfair writing, you will be unable to unlock the dialog choices necessary to learn Force Sight without either losing influence with Visas or picking up some Light Side points. Since you can't avoid this conundrum, your best bet is to take the alignment hit until you learn the Force Power and then choose the Dark Side options on your way back out to make up and Dark Side points that you lose. Wrap up any loose ends on Onderon/Dxun and head back to the Ebon Hawk.
Handmaiden - Influence Increase - By this point you should be at level 19 or higher. Go ahead and initiate the final sparring match with Handmaiden. Afterwards you will be able to train her as a Dark Jedi Guardian.
NOTE: The only influence opportunities of any real significance revolve around Disciple, the party member that female PCs will receive in place of Handmaiden. Luckily you can make up for "lost time" by maxing out your influcence and training him to become a Jedi before you leave the room you found him in.
Disciple - Influence Decrease - When you first speak to Disciple after he joins your party, he will comment on your Dark Side alignment. No matter what you say, you will gain an automatic influence decrease.
Speak to Disciple again, any topic will suffice. After you are finished and attempt to leave, he will stop you to ask about your exile. When he asks if you have a record of the trial, tell him that you do, however do not agree to let him see them. This will prohibit you from inadvertently picking up any influcence increases.
Once again, speak to Disciple. Despite your previous exchange, he will admit that he has seen the holorecord of your trial. Before you end the dialog, ask him what he does. He will comment that he has some medical training. Ask to be examined and after his has finished his prognosis, select "Save your counseling for the weak - I feel powerful, strong."
Initiate another dialog with Disciple and ask him what he does for the Republic. When he comments that he is trying to perserve the Jedi histories, ask him why he feels that is important. After he admits that he has tried to keep an open mind about the Jedi, select "It is the truth - the foolishness and arrogance of the Jedi has brought much harm to the galaxy."
After he admits that he has ofter felt the same way, keep the conversation going, inquiring about the various key planets in the Republic. Ask him "How did this happen?", and then follow it up with "So what? Does the Republic deserve to be saved?". When available, begin wrapping up the conversation by stating "The galaxy belongs to those who can hold it. Perhaps the Sith are a better choice."
Talk to Disciple again and comment "You look familiar to me.". He will admit that you two knew each other before the Mandalorian Wars. When he laments his missed opportunity of Jedihood, respond with "It is good you turned from the weakness and hypocrisy of the Jedi - as I did."
Speak to Disciple one last time. Ask him about his Jedi training on Dantooine. After he finishes relaying the story, tell him "Well, I doubt you could have completed the training anyway." Follow this up by offering to train him. He will accept, becoming a Jedi Consular.