Question

Why did the Byzantine Empire become more Greek than Roman?

Answer 1

        Byzantines didn't "become more Greek than Roman". They were already Greek to start with. 
        But anyways... lets not have me rant about that part.
  The population was Greek. The empire was split into West and East. West centered on Rome (before the fall) and the East on Constantinople (A town in Greek). From the division working language was Greek.
  Greek language and couture was more prevalent in the Eastern Roman Empire. When the fall of the West Byzantine court held onto the Latin language for official decrees and other state documents, but eventually even that was lost to the grater Greek influence.
Hoped this helped! Remember to change this to your own words. <span />