As with many things relating to cultural distinctions and identification, the transition from Roman (Ῥωμαῖοι) to Hellene (Ἕλληνες) was a long process.

The process arguably starts in the 10th Century, by this time paganism was no longer an existential threat the Christianity, and the Empire of the Romans witnessed a renaissance of writing and art as the Byzantine Dark Ages came to an end. As such the term "Hellene" went back to it's original secular meaning. That being a person from Greece who speaks Greek, and not a believer in the Pagan Gods, as the term had been associated with during Late Antiquity. The Hellenic identity would also start to be accelerated thanks to the arrivial of the Crusaders, and increasing anti-Latin sentiment in the Empire, which culminated in the Massacre of the Latins in 1182. In the 13th century, following the partition of the Eastern Roman Empire after the 4th Crusade this anti Latin sentiment became even stronger. The Crusaders had sacked Constantinople and destroyed the Empire, creating a "Latin Empire" Crusader state in it's place. From 1204-1261 there was no longer a single Roman state in the area and the title was claimed by multiple native and non-native states. Such as the Latin Empire, Nicaea, Eprius and Trebizond.

It's probably no coincidence that the Hellenic identity saw a strong resurgence during these years, as the absence of a single Roman state weakened the legitimacy of continuity with the Caesars and the Greeks not under Latin rule undertook a mass resistance campaign against the Latin Empire, with the goal of returning Orthodoxy to Constantinople.

To give some examples, Theodore Alanias at the time wrote quite strikingly his brother that "the homeland may have been captured, but Hellas still exists within every wise man" and the 3rd Nicaean Emperor (Theodore II Laskaris) went further, saying: "the Hellenic race looms over all other languages. Every kind of philosophy and form of knowledge is a discovery of Hellenes... What do you, O Italian, have to display?"

Despite Hellenism becoming fashionable again during this time, it did not replace the "Roman" identity entirely. The Greeks still saw their nationality as Roman, and only Greek scholars/intellectuals started to think of themselves as part of the Hellenic people. However even among the elite this wasn't universal. The last Roman Emperor, Constantine XI, upon his coronation at Mystras in 1449 was asked by some of his Italian advisors to take the title "Emperor of the Greeks" rather than "Emperor of the Romans" to avoid offending Italian/Papal sensibilities. However Constantine refused this offer, taking the tradtional title of "Emperor of the Romans" upon his coronation. This suggestion however seems to have stuck with him throughout the rest of his reign, as he would refer to his people during the Siege of Constantinople as "Greeks and Romans", pandering to the aspirations of both sides. Among the rural population Roman was still the dominant identification throughout the Byzantine and Ottoman periods.

The Ottomans would also continue using the term Roman (Turkish: Rûm). Although to the Ottomans it wasn't exclusively used for Greeks, but for the Orthodox population of their Empire. Christian Bulgarians, Serbians and Greeks were all Rûm. Yannis Makriyannis, general in the Greek War of Independence for example, recalled in his memoirs something one of his friends had told him:

"What say you, is the Roman State far away from coming? Are we to sleep with the Turks and awaken with the Romans?"

During the Age of Enlightenment there was a growing romanticism with Ancient Greece in Western Europe, and over time this also spread to the Ottoman Empire where many Greek intellectuals in the Ottoman Empire began to advocate for a return to the glory of classical Hellas. When the Greek War of Independence began the Hellenic identification gains considerable traction among the Greek rebels. Roman was what the Ottomans called them, whereas Hellene was what the Greek patriots and intellectuals called themselves. Because of this Roman came to be associated with "Pro-Ottoman" or "Slave" where as Hellene became associated with "Freedom Fighter" or "Patriot". The obedient, enslaved Greek answered to Roman, the patriotic Greek answered only to Hellene.

However the Greeks only managed to carve out a small kingdom in Attica and the Morea during the war, as a result most Greeks still remained Ottoman subjects. These Ottoman Greeks in Macedonia, Thrace and Asia Minor, unexposed to the Hellenic nationalism in Athens retained their Roman identity. Only when these Greeks were made part of the Kingdom of Greece and given Greek education did they adopt the Hellenic identity.

In short, there is no single point in which we can say the idea of being Greek overtook the idea of being Roman. However there's alot of important events that took place during this process which gives one a good idea of how and why this took place. In some ways the Roman identity still survives today as even in modern Turkish, the tiny remaining Greek diaspora in Turkey are called Rûm, and some Turkish Greeks choose to identify as such.

If you have any further questions, or want me to further elaborate on some of my points then feel free to do so. This is a rather long and complicated subject so if you still find some things unclear then just ask.