The nature of the question needs further clarification.
It should be noted that the phrase..."Greek speaking Romans", is misleading and historically inaccurate.
The Byzantine Empire, during its earliest years, were governed by a mixture of Greek speaking Greeks, Romans, as well as a few Illyrians-(present-day Albanians). However, the Byzantine peoples, that is to say, the indigenous Byzantine civilian population/the centuries old residents of Byzantium proper-(later Constantinople), were overwhelmingly, Greek speaking Greeks, with a much smaller number of bilingual speaking Romans-(who were equally fluent in Latin and Greek). Latin, during the early years of the Empire, was probably used conversationally and textually by the Byzantine Greeks, though it had parenthetical value when compared with the use of their indigenous Greek tongue....and script.
As the Byzantine empire aged over the centuries, the Roman or Latin aspect of the Empire ended and beginning around the 600's AD/CE, the Byzantine Empire would emerge as an exclusively Medieval Greek Eastern rite Christian Empire-(both in terms of its ethnic governance and civilian population). The empire's non-Hellenic subjects/ residents, were never referred to as, "Byzantines" per se. The Greco-Byzantines, similar to their Ancient forefathers, did use rather disparaging and pejorative sounding words to refer to the non-Hellenic residents of the empire by labeling them as, "barbarians" and even established a rather notorious and discriminatory, "Bureau of Barbarian Affairs".
As for the Greco-Byzantines being aware of the earlier Roman civilization, they absolutely were aware of it. Remember, the Byzantine Empire, while predominantly Hellenic in character and history, did often identify themselves as, "Romans" in terms of their political-(though not their genealogical, nor rhetorical) legacy.
Having said that, the Aristocratic, Educational, Clerical and Political elites, had direct access to the literatures and writings of Rome. The earlier Roman writings were preserved-(along with earlier Greek writings), at The Library of Constantinople.
As to whether or not the average Greco-Byzantine civilian had an extensive or a moderate knowledge of Roman writings and history is somewhat questionable. However, having lived under pagan (pre-Christian) Roman rule for 400 plus years, it is very likely that the average Greco-Byzantine resident had, at the very least, a peripheral familiarity with and working knowledge of....Rome's historical and textual legacy.