They would think of themselves as Roman, but not Latin. Depends on where they lived, but mostly the Greeks maintained their linguistic majority in their homeland, and the educated Romans spoke Greek as their lingua franca. Greek slaves were prized, as the Romans felt a sense of cultural fraternity with the Greeks. The situation becomes even more interesting when the West falls. Not only did the eastern Roman empire become essentially a Greek empire (in that the state switched from Latin to Greek, and moved away from Rome's ecclesiastical authority culminating in the schism of 1054. Even so, they still called themselves romans, and it was contemporarily lauded as the end of the Roman empire, as the Byzantines were still thought of as romans, and the word Byzantine wasn't even coined until the 1800's.