The answer is generally "both... with a healthy admixture of other peoples, as well," but it depends on when you're asking about! Moreover, keep in mind that most genealogical talk in Byzantine sources comes from discussions of imperial and aristocratic families, who would have had more of a vested interest in portraying themselves as of Roman stock than the average "guy on the street."

(Incidentally, I discussed the Hellenes/Rhomaioi question in this post a few months ago—while the latter was certainly more common in Byzantine texts, it's not strictly true that hellenes was only used to describe pagans and heretics. We see the word applied as a self-designation in a number of texts, especially following the Fourth Crusade.)

A few examples here might make what I mean more clear. First, the "Macedonian dynasty" of Basil I (r. 867-886) and his descendants claimed descent from Alexander the Great, Constantine I, the Arsacid dynasty of the Parthians (via an Armenian branch), and the Persian emperor Artaxerxes I. (A recent article by Nathan Leidholm suggests that while this last claim may be an extreme outlier in terms of Byzantine royal ideology, we should see it as part of a longer tradition of Persian and Hellenistic claims to dual Achaemenid/Arsacid descent.) Constantine was a particularly popular focus of genealogical claims for obvious reasons, and appears as an ancestor of the Doukas (who are also described as of Italian origin in the satirical dialogue Timarion) and Phokas families, among others. The eleventh-century historian Michael Attaleites even claimed that his patron, the emperor Nikephoros III Botaneiates was descended from the Roman Fabii (as well as Constantine, of course, through the Phokades). The (Byzantine) Serbliai rather simply associated themselves with the (Roman) Servilii, and a twelfth-century lead seal explicitly proclaims the "Roman" heritage [ῥίζαν γένους ... 'Ρώμης] of one John Mankaphas.

At the same time, we see the criticism—also from the twelfth century—by Michael Italikos of "the stories tracing families to Peleus and Ajax, finally attach the family line to Zeus," and, as I mentioned above, a noticeable uptick in self-designation as hellenes after 1204. This was not, however, always seen as mutually exclusive with claims to Roman heritage. Perhaps the fourteenth/fifteenth-century scholar Manuel Chrysoloras put it best in a 1414 epistle to then-emperor Manuel II Palaiologos:

Let us remember from what men we are descended. If someone would like, he could say that we descended from the first and age-old, I mean from the most venerable and ancient Hellenes (no one has remained ignorant of their power and wisdom). If you please, you could also say that we descended from those who came after them, the ancient Romans, after whom we are named and who we are now named and who we, I suppose, claim to be, so that we even almost erased our ancient name. Rather both of these races came together in our times, I think, and whether someone calls us Hellenes or Romans, that is what we are, and we safeguard the succession of Alexander and that of those after him.

But Chrysoloras's compromise between Greek and Roman identities, like the Council of Florence which was soon to follow, would not be a lasting one. In the following generations, Greek scholars—especially those who fled to Latin Europe (where a Roman identity would be more contested) after the fall of Constantinople—clung ever tighter to their Hellenic identity.

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Bibliographic note—in addition to the works cited in the linked answer, see:

Leidholm, Nathan. "Political Families in Byzantium: The Social and Cultural Significance of the Genos as Kin Group, C. 900-1150." Ph.D., University of Chicago, 2016.

---. "Nikephoros III Botaneiates, the Phokades, and the Fabii: Embellished Genealogies and Contested Kinship in Eleventh-Century Byzantium." Byzantine and Modern Greek Studies 42, no. 2 (2018): 185-201.

---. "Artaxerxes in Constantinople: Basil I’s Genealogy and Byzantine Historical Memory of the Achaemenid Persians." Greek, Roman, and Byzantine Studies 60 (2020): 444-71.

Steiris, Georgios. "History and Religion as Sources of Hellenic Identity in Late Byzantium and the Post-Byzantine Era." Genealogy 4, no. 1 (2020): 16-32. [link]

This is a question that exists on a gradient. The Byzantine Empire consisted of several different cultures and religions that identified in different ways. However, the Byzantine elite, and certainly its emperors, did largely consider themselves as Roman, not Greek. The Byzantine rulers insisted on being called Emperor of the Romans. How much the distinction mattered to any individual (especially those outside of the aristocracy), based on their culture, personal history, and etc. is a discussion that is much more open to debate. Any individual person could trace their lineage back to any given point of origin. However, what mattered was whether or not that point of origin was Roman. Most Byzantines spoke Greek and had Greek, Balkan, or Anatolian descent. However, to them, these things made them Roman, as these regions were all a part of the Roman Empire. They might not have thought of themselves as literally Italian, that depended on how far back someone might trace their ancestors, but definitely as Roman.

It is worth keeping in mind that, even with the "Fall of Rome" in 476, the people of the time did not see this as the collapse of the Roman Empire - it was believed to be continuing on just fine in the east. To the people of Byzantium, this represented no de jure change. They were the same empire, governed by the same class of people, ruled by the same Roman Emperor. While the empire might have been split into eastern and western portions, it was still "The Roman Empire", and losing half of it did not mean that the entire empire collapsed. Even categorizing the west as having been "lost" is somewhat of a misnomer. Odoacer, who deposed the last western emperor Romulus Augustulus, paid nominal fealty to Emperor Zeno in Constantinople. While Odoacer is called king in many documents, and certainly wielded power in Italy, he maintained the appearance of fealty to Rome, and service to the Roman Emperor. He often referred to himself as patrician, a Roman title, and made appointments to Roman legislative bodies, and was said to be ruling Italy on Zeno's behalf. Odoacer even took the imperial regalia and sent it to Emperor Zeno in Constantinople.

While Odoacer's conquest was certainly the end of Roman hard power in the west, the legacy continued. The theater of Odoacer's service to Rome was important for his legitimacy, but also propagated the belief in Byzantium that the Roman Empire certainly had not fallen. After all, Zeno ruled from Constantinople, and Odoacer governed the west on his behalf. They were still Rome - because they perceived no fall until much later, they had no concept or reason to reevaluate their opinion. Odoacer's successors, including Theodoric, followed a similar layer of theater. Even after hundreds of years and the decline of the Byzantine Empire, the aristocracy certainly never let go of this claim and political theater; while they did eventually acquiesce to being addressed by Western European dignitaries as something other than Rome (due to the competing claim of the Holy Roman Empire), they never relented on claims that they were the Roman Empire.

Overall, even though the Western Roman Empire de facto fell in 476 and Odoacer seized effective power in the region, the Byzantines did not view this as the collapse of the west. They viewed the west as being governed on their behalf, and thus, they did not perceive Rome having fallen. Eventually, the Byzantine Empire would decline and their soft power in the west would also diminish, and rulers of Italy would slowly cease at playing theater in giving service or fealty to the emperor in Constantinople, but the Byzantines themselves never ceased in the de jure insistence that they were, in fact, the Roman Empire. They had good cause to believe such for a long time.

Bibliography:

The Fall of the Roman Empire: How and Why It Came to an End is a wonderfully educational chapter in a book called Introduction to the History of Christianity by Richard A. Todd

History of the Later Roman Empire: From the Death of Theodosius I to the Death of Justinian by J.B. Bury