The Byzantines viewed their pagan Roman ancestors as the direct progenitors of their own state and time. They obviously denied the divinity of the old emperors and many of the buildings commissioned in pagan times, and even long before the Roman empire existed (the Parthenon was made into a shrine dedicated to Mary for example), were turned into Christian sites, but the Byzantines sat at a weird position of both Christian and Classical tradition.

I wrote about this a bit more in an answer on the Trojan/Roman reception in Byzantium and I'll repost it below:

Quite simply, the Byzantine viewed themselves as he legitimate continuity of the Roman state as instituted by Romulus, descendant of Aeneas, Prince of Troy. Anthony Kaldellis, one of the preeminent scholars of Byzantium has this to say on the topic:

As Christians, the Byzantines traced the grounds of their community to the Chosen People of the Old Testament, but as a nation their consciousness was thoroughly Roman. Institutional continuity and excellent historical records easily established the authentic Roman origin and imperial legitimacy of their state and society. Nor was this done defensively, as though anyone seriously doubted it. In the sixth century, the emperor Justinian traced the ‘‘ancient history of the government’’ back to ‘‘Aeneas, the King of Troy, Prince of the Republic, from whom we are said to descend,’’ adding that ‘‘Romulus and Numa founded the government.’’

So the short answer to your question is that the Byzantines viewed themselves as the inheritors of the decidedly Roman tradition over their Greek location. Indeed, professor Kaldellis has described the medieval Roman Empire as a proto-nation state, in that it encompassed an identity, Roman, that was tied to cultural continuity with the Roman state that at the time was Greek speaking and followed the patriarch of Constantinople in religious matters. Everything else was more or less negotiable. Indeed, according to his interpretation the Bzyantines were not so much as Greekified Roman Empire as they were a Roman Empire that currently was dominated by Greek speakers due to its, temporary one assumes, loss of lands in the west.

Now this is not to say that the Byzantines rejected their Greek heritage and all that it entailed, but that they chose to, usually, highlight their continuities with the Roman past for their political, religious, and cultural authority. The Byzantines thus were the heir to the Roman political/mythological history that traced their descent, ultimately, to Troy. The status of the majority of Byzantines as Greeks as well was not as important to their identity and presentation. They were Romans living in Greece, spoke Greek, and read ancient Greek authors (or more often Greek Church fathers), but they were still Romans.

Now this gets significantly more complicated when you're talking about the Latins/Franks (the terms were often used interchangeably and in reference to any western European really) and how they viewed Byzantium, because it was, in their eyes, decidedly Greek with all that entailed, namely negative things and a rejection of their authority as Roman Emperors, but that is an entirely different question than the one that you asked.