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.
Hali ya swali hilo inahitaji kufafanuliwa zaidi. Ni lazima ieleweke kwamba usemi..." Waroma wanaozungumza Kigiriki", ni wenye kupotosha na usio sahihi kihistoria. Milki ya Byzantium, wakati wa miaka yake ya mapema, ilitawaliwa na mchanganyiko wa Wagiriki wanaozungumza Kigiriki, Waroma, na vilevile Wailiria wachache (Walbania wa leo). Hata hivyo, watu wa Byzantium, yaani, wenyeji wa Byzantium raia idadi ya watu wa karne za zamani wa Byzantium sahihi (baadaye Constantinople), walikuwa wengi, Kigiriki kuzungumza Wagiriki, na idadi ndogo sana ya lugha mbili kuzungumza Waroma (ambao walikuwa sawa fluent katika Kilatini na Kigiriki). Katika miaka ya mapema ya Milki ya Roma, Kilatini kilitumiwa kwa mazungumzo na maandishi na Wagiriki wa Byzantium, ingawa kilikuwa na thamani ya kipekee ikilinganishwa na matumizi ya lugha yao ya asili ya Kigiriki na maandishi. Wakati milki ya Byzantium ilipoanza kuimarika, sehemu ya Kirumi ya Milki hiyo ilikoma na kuanzia karibu 600 KK, Milki ya Byzantium ilionekana kama Milki ya Kikristo ya ibada ya Mashariki ya Kigiriki ya Enzi za Kati (kwa suala la utawala wake wa kikabila na idadi ya raia). Raia wasio Waekrania wa milki hiyo, hawakuwahi kutajwa kuwa, "Wabyzantium" wenyewe. "Wagiriki wa Byzantium, kama walivyofanya mababu zao wa kale, walitumia maneno yenye kudharau na ya kudharau kwa kuwahusu wakazi wasio Wagiriki wa milki hiyo kwa kuwaita ""Wagharibu"" na hata kuanzisha ""Bureau of Barbarian Affairs"" yenye sifa mbaya na ya ubaguzi." Kwa sababu ya kuwa na utajiri wa Greco-Byzantine, watu wa kale wa Roma walianza kuamini kuwa kuna Mungu. "Kwa mfano, wakati wa utawala wa Byzantium, watu wa Roma waliitwa ""Warumi"" kwa sababu ya asili yao ya kisiasa, ingawa si kwa sababu ya ukoo wao." Baada ya kusema kwamba, aristocratic, Elimu, Clerical na kisiasa wasomi, alikuwa na upatikanaji wa moja kwa moja kwa fasihi na maandishi ya Roma. Maandishi ya kale ya Kiroma yalihifadhiwa - (pamoja na maandishi ya kale ya Kigiriki), katika Maktaba ya Constantinople. Kama vile au la raia wa kawaida wa Ugiriki na Byzantium alikuwa na maarifa makubwa au ya wastani ya maandishi ya Kiroma na historia ni jambo linaloweza kuhojiwa. Hata hivyo, baada ya kuishi chini ya utawala wa kipagani (kabla ya Ukristo) wa Kirumi kwa zaidi ya miaka 400, ni uwezekano mkubwa kwamba mkaaji wa kawaida wa Ugiriki-Byzantium alikuwa, angalau, na ujuzi wa kijijini na ujuzi wa kazi ya urithi wa kihistoria na maandishi ya Roma.