I'm currently playing a fairly successful game that I started as doux in Trebizond. My liege lost the kingdom of Armenia to the Seljuks but they fall to infighting fairly often and I took the chance to expand into parts of Georgia and Armenia. As for expanding within the Empire try to marry other lords for claims and allies because every time I've fought another doux they've brought help. Also if you hold more than one duchy other doux will conspire, eventually asking the Emperor to join in and cut you down to size.
It can be frustrating playing as doux in Byzantine because it gets targeted by for Jihads a lot and if you don't have a very good liege you can get conquered no matter what you do. I ended up marrying the Emperor's sister and after he lost Greece (in addition to Armenia which was lost earlier) I started a faction to place my wife on the throne. Right now I'm playing as the son who is now Emperor and attempting to fix things.
