I think neither roman or greek adjectives are relevant for Byzantines : they were byzantine ^^. But this is the "a bit too simple answer" and I'll try to get something more relevant.
This question mainly comes from the views and the ideas foreigners to the Byzantine Empire we use to create its history. Emperors of Constantinople were of course seen, and wanted to be seen, as Roman Emperors, and their foreign policy had always been to act as if they were (attempts to reconquer the former possessions of the Roman Empire in Spain against Wisigoths, Africa against Vandals and Muslims, Levant, South Italia...). Another evidence is the name Seldjuks gave to their conquests on the Byzantines : "Sultanat of Rum". But I would add that Europeans often used the Roman Empire ideology for the Byzantines when they wanted to pleased them, or to be recognized. When things were going worse between Westerners and Byzantines, I would here agree with Plotinus, the Hellenistic approch prevailed and was often used with pejorative purposes. Hellenistic culture indeed includes oriental influences, which was seen as decadence or futility. (I would advice to read Liutprand's ambassades reports for both admirative and despiteful sights)
Now, if we consider the Byzantine opinion about themselves, it requires two parts : one for the high-level byzantine social group (let's call this aristocracy), and another for the common people.
I think we can trust the aristocracy to have strongly believed in their roman heirloom ; being Christians, there still were some roman customs actively practiced in power, administration and military domains : its name "Basileia Rômaiôn", a lesser cult of the Emperor (though they were Christians, the byzantine Emperor is THE exemple of a thaumaturge king), legions, the ideology of universal monarch (which will effectively disappear, fighted by Ghalifa and Holy Roman Empire, both of them reclaiming it too), the idea of the triumphant general as emperor. This last one is very interesting, as many byzantine emperors were victoious generals. The Roman Empire didn't use lineage successions as the Byzantines were theorically doing, and often, Byzantines have accepted both male primogeniture and this ancient notion to choose their leader (Nikephoros II Phokas, John I Tzimiskes or Michel IV the Paphlagonian are good examples). On the other hand, we cannot deny to christianism its powerful influence on their culture, nor the fact that Rome in Greece and Asia has never actually been Rome, hellenism has been too strong not for the Romans to be acculturated there ; they even had been in Latium. So the more I can say is that byzantine elites were "hellenistic people with a mixture of christian and ancient roman ideology".
Common people are more difficult to determine, as they left less testimonies of how they were feeling about all that and I must say that I don't know very much about them... The base of the byzantine economy was agrarian, and the population was regrouped around manors (aristocratic manors or possessions of monateries) and villages. Villages were communal free entities (koinotes) which included the inhabitations (oiketores) and the surrounding fields (kôrion). In the manors, the workers were mainly parekos (some kind of serfs if I understood well). These last facts tend to make me think the people were more "Greeks" than Romans, but I may greatly mistake on that one, because I'm only influenced by the similarities between oiketores/kôrion and ancient asty/kôra, and by a bit of bad teleology : after all, the Greeks of the XVIIIth century were more greek than roman (and of course, christian)...
Conclusion : if you're considering Byzantium as a state, I would say "Greek" is not the right term, "Roman-Hellenistic" would be better. Then, if you're considering Byzantium as a people : Hellenistic and Christian. Now, I would like to add that "Greek" isn't a relevant term only because we don't use it during the whole Middle Age, which is quite confusing : of course inhabitants of Greece and Asia Minor are not Ancient Greeks anymore, their culture has evolved, influenced by other ones, but I daresay that's the same with every culture at anytime of history... so if you were asking this question in a matter of continuity, the answer is that they were approximatively Greeks, in the same way we can say Angles were British, or Franks French... culture's a everchanging conceptual thing, with variation between people of the same group, so you can never be 100% right when trying to put a name on it ^^.