Full solution (if you wish to do female romance)
----
Keep talking to Carth, after every level up on Taris. And after every star map. After every star map, return to Dantooine with Carth. You'll meet one of his friends and you will be on your way to Korriban.
On Korriban, take Carth (naturally) and when you go through the quarters on Korriban, you'll meet his son. You'll have to prove to him that the sith are the bad guys. If you can't hack the door on Uthars room then go to Yuthura and she'll tell you her plan to kill Uthar. Now, go to Uthar and tell him. Go back to Yuthura and tell her that Uthar has some how found out. She'll give you the key and you can break in Uthars quarters. In the footlocker, you should find a database. Go back to Dustil and that should be it. :cool:
WARNING: If you do not complete the quest after meeting Carths friend, you cannot proceed in the romance.
If your male, it won't matter when you find the friend, just make sure its before the Leviathon.