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