The lingua franca and administrative language was Greek. The Empire called itself Roman, but its scholars knew a lot about Ancient Greek and very little about Rome. The core of the Empire was Asia Minor, much of which was Greek-speaking until the Turkish invasions, and Greece.
But at least until the Fourth Crusade, the Empire was bigger than Asia Minor and Greece, and its ethnicities matched—whatever language they may have used in writing and the church. A large number of its emperors and officials were ethnic Armenians, and had the surnames to match (Category:Armenian Byzantine emperors, e.g. John I Tzimiskes). The scholar John Tzetzes was proud of his Georgian heritage. John Koukouzelis, who reformed Byzantine chant, was most likely Bulgarian.
I have a cousin with an Arvanitika surname, Tzathas. When she went to uni, her prof commented how her surname sounded “Byzantine”. And it would indeed sound Byzantine, since that was when half the surnames of famous people weren’t Hellenic.
Byzantines resented Westerners calling them the Greek Empire instead of the Roman Empire—not only because it denied them continuity with the Roman Empire, but also because Byzantium wasn’t confined to Greece. Of course, Byzantines, and their successor state the Ottomans, were profoundly indifferent to ethnicity anyway, which is why the non-Hellenic background of Byzantines was not paid much attention. What mattered was religion; which is why the Ottoman Rum Millet and the Modern Greek use of Romioi (Names of the Greeks ) primarily referred not to ethnic Greek, but to Orthodox Christians.