In 1453 Constantine XI Palaeologus died as Basilèustṑn rōmàiōn (emperor of the Romans) while defending the capital of the Basilèiā Rhōmàiōn (empire of the Romans). Having conquered Constantinople, the Ottoman sultans called themselves, among other titles, qaysar-ı Rum (Caesar of the Romans) in order to be recognised as sovereigns by the Rhōmàiōn (Romans) who inhabited the Greek and Anatolian peninsula.

The so-called Byzantines never stopped considering themselves Romans until after the fall of Constantinople, and in any case 'Byzantine' remained a term referring to the ancient Greek city-state of Byzantium until the 18th century. This is not to say that they did not also consider themselves 'Hellenes'. The identity of the Eastern Romans over the centuries became a fusion of the Greek and Roman elements, without ever feeling the need to deny one or the other.

There's a great book on this exact subject by Anthony Kaldellis (Hellenism in Byzantium).

To summarize Kaldellis' book (although I highly recommend anyone interested in this read it):

Hellenism as an identity pretty much ceased already before the 5th Century A.D as the Roman identity supplanted it in almost all areas. By 400 A.D most Greek speakers would have seen themselves as Romans, not Hellenes. As Kaldellis states:

Between Synesios and Psellos, so for a period of six centuries, Hellenic identity went into abeyance. It had only a hypothetical existence, being a relic of the past that could be glimpsed in ancient texts; or the antithesis of Christianity, something that could both negate and complement but whose power was never actualized; or fragmented markers that signified little, for example the mere fact of language or geography

The Romanization of the Eastern provinces, combined with the spread of Christianity (and thus, the association of "Hellene" with "Pagan") put the Hellenic identity in a sort of limbo for several centuries. You barely hear anything about Hellenes during this time other than vague historical/religious allusions or geographic terms.

When the empire organized its military districts (themes) in the 7th Century, there of course was a 'Theme of Hellas', but this was only on the same level as say, the 'Thracesion Theme' or the 'Opsikion Theme. While said Themes would occasionally be compared and contrasted with eachother, these were thoroughly sub-national identities (if such a term is even appropriate for the period).

So dead was the Hellenic identity up to the 11th Century that the Hellenes were often described as a foreign people. Greek speakers of the 8th Century would not view themselves as being in some 'Hellenic ethos'. Hellenes were a long gone, foreign people of different faith, much like Egyptians or Persians. There was no sense that the Hellenes of old still existed, nor was there much interest in their implied demise. The only place you could find occasional, poetic use of the term was for people well-versed in the Ancient Greek classics, who would "Hellenize their tongue". In this sense, Hellenism was, like Kaldellis says, a trained skill and not a national/cultural identity.

So as we have established, the old Greek identity was entirely dead in the earlier centuries of Byzantine history. However, starting in the 11th Century there is a slow, very gradual revival of the term.

It starts most prominently with Michael Psellos. Psellos was one of the most learned men of his time and deeply invested in both theological and Platonist philosophy, to such an extent that his contemporaries even doubted his faith. To Psellos, Pagan myths should not just be dismissed off-hand as foolish stories, but should be studied. For Psellos, the Pagan texts and rites contained 'hidden meanings' that could be understood/useful from a humanistic, Christian context.

While Psellos still understood himself throughly as a Roman, and cites 'historical Romans' such as Romulus on several occasions, it is with Psellos that the Hellenic identity starts re-appearing. In his Chronographia, Psellos boasts that he can't find:

even a seed of wisdom in Greece or the barbarian lands, though I searched everywhere

Here, for the first time in a long time, Psellos calls not just a geographic region, but his whole empire/nation 'Greece'. This is likely just an intentional anachronism, to contrast the 'barbarians' with the ancient 'Hellas' his like-minded scholars would recognize and have vivid images of. But it is still an interesting development.

This development would continue in the subsequent 12th Century, as there would be a new growth in more humanist-oriented Greek literature. Like with Psellos, none of these writers probably had the intention of reviving or promoting a Hellenic identity, but the promotion of the ideas would contribute to later developments in the 13th Century.

It's during the 13th Century that you really start to see the Hellenic identity properly re-emerge among segments of the learned elite.

The 13th Century was turbulent for the empire. The century started with the sack of the capital and the partition of the empire among Crusaders. Such a catastrophic series of events led to a re-thinking of many things that previous Byzantine writers avoided talking about. Writers of the time must have been all too aware that the city from which their name derives from lay in the west, and that the figures from which they claimed descent from, were from Italy. Which must have been a very difficult and painful topic to wrestle with given that Italians, which they were actively resisting, had just sacked their capital and destroyed their empire. The Roman identity had become destabilized, and difficult to grasp in as concrete ways as before.

Because of this, Hellenic identity started becoming more popular among intellectuals, especially when contrasting themselves against the Latin Crusaders. Hellenic heroism or high culture will be contrasted against the barbarous and tyrannical Latins.

During the imperial exile (1204-1261), the term "Roman" also begins to become more nebulous. It referred to the Romans themselves, but could also refer to the Catholic Church/Pope, 'Roman' in the texts become increasingly dependent on context.

Point being, Hellenes are no longer just someone who has mastered the ancient classics, but now can refer to anyone who speaks Greek.

This reaches its boiling point during the reigns of Emperors John III Vatatzes and Theodore II Laskaris. John III, who recovered vast amounts of territory from the Latins, writes the following in a polemical letter to Pope Gregory IX in 1237:

Ioannes Doukas, faithful-in-Christ basileus and autokrator of the Romans, to the Most Holy pope of elder Rome, Gregorios . . . When those who were sent by your Holiness approached my Imperial Majesty they gave me a letter, which they claimed was yours and insisted that it was addressed to me. Yet I, seeing that its contents were absurd, could not believe that it was yours and thought that it was by someone who is extremely irrational, whose soul is full of delusion and arrogance . . . This letter says that wisdom reigns in our Hellenic race and streams of it flowed out to all other places as from a spring;

[...]

That wisdom springs from our genos and that it blossomed first among us before being transmitted to others . . . is said truly. But how did you forget, or, rather, if you did not forget, how did you suppress the fact that, in addition to our reigning wisdom, imperial authority in this world was also bestowed upon our genos by Constantine the Great?

Now, this is not John III claiming the Hellenic legacy, on the contrary this letter is him re-affirming his Roman legacy/legitimacy, and disparaging the illegitimate ways the Pope/Latin Empire claimed their claim to the Roman legacy.

However John does this in an interesting and new way. Instead of viewing the Hellenes as some old, extinct people which the Romans replaced, now the Romans are also Greeks. That's to say, the Roman legacy has passed from the Latins to their "genos". Thereby Hellene and Roman are now almost interchangeable, though Roman still takes precedence.

During the following centuries of the empires existence, this interchangeable use of 'Hellene' and 'Roman' became the status quo, which is radically different from how the terms would have been viewed before the 13th Century. Some scholars, such as Plethon would even go so far as to advocate for rejecting Christianity and returning to the Hellenic gods.

Though it should be noted that this was very likely only the case among the well educated elite of the empire. For the common folk, these identities would not have the loaded and philosophical meanings you see here, and Roman was still the dominant identity. Ottoman Greeks almost universally identified as Romans, and their Turkish rulers called them such. The Hellenic identity does not begin to fully replace the Roman identity among ordinary people until the Greek War of Independence in the 19th Century and the subsequent incorporation of the vast majority of the Ottoman Empire's Greek speaking territories into the modern Greek state.

E.g. Did the late Palaiologos dynasty see themselves as Roman emperors?