I'm not going to say everything about how she reveals her name, as that is just ruining some of the side-quest of finding out about her.
However, the main thing you need, and a lot of with Handmaiden, is influence. You get influence by fighting her whenever you get the chance on the Ebon Hawk, and when you win, you will gain influence from her, which sometimes opens ups new dialogue options.
Also, to quote Battlewookiee:
P.S. Shouldn't this be in the spoilers forum....Battlecookiee?