Step 1: In the early 19th century, Russia takes over a bit more of the Caucasus than they did in OTL. They create a separate Kingdom of Georgia, Kingdom of Armenia, and Kingdom of Greece. These are in personal union with Russia, but have separate governments. Together, the union of these four kingdoms is known as the "Roman Empire", demonstrating both Russia's claim to be the Third Rome and the fact that many Greek people at this time consider themselves Romans. The term "Roman Empire" is not widely used in other countries.
Trebizond is named the "temporary capital" of the Kingdom of Greece, with the expectation that Constantinople will eventually be retaken.
Step 2: Russia and its puppet kingdoms of Greece and Armenia steadily take over more and more of Anatolia from the weakening Ottomans. Most Turks from the conquered parts of Anatolia migrate to the remaining Ottoman territories, particularly Turkey-in-Europe. Anatolia becomes fairly depopulated as a result, but this is offset by Russian colonists arriving to replace the expelled Turks.
Step 3: Eventually all of Anatolia is taken over by the Roman Empire/Kingdom of Greece, as are the Greek islands and Pelopennese. The Turkish population in Europe is far higher than in OTL as a result of this. The Ottoman Empire is overthrown by Turkish nationalists and a Turkish nation-state is established in Europe.
The Roman Empire/Kingdom of Greece tries to cross the Bosporus and take Constantinople from the Turks, but they are unable to do so. In the end the new Turkish Republic reaches a peace agreement with the Roman Empire, and a population transfer between the two countries is carried out.
Step 4: A communist revolution takes over Russia in a similar manner to in OTL. The four kingdoms of the Roman Empire (Russia, Georgia, Armenia, Greece) are officially merged into a single united Roman Empire. Large numbers of anti-communist Russians flee to the Roman Empire.
Current situation: The Roman Empire exists with much the same borders. Few people use the term "Greece" any more; indeed, the term "Hellenic" is sometimes considered an insult used to downplay the Roman identity. The Roman Empire strongly considers itself to be the direct continuation of the original Roman Empire. It's pretty socially conservative and theocratic.
The Turkish capital is Istanbul, on the western side of the Bosporus. The Roman capital is Constantinople, on the eastern side of the Bosporus. The two countries glare menacingly at each other.
The primary language of the Roman Empire is Greek, although there's a large Armenian minority in the east, and large Russian and Bulgarian minorities in central Anatolia (Russians mostly descended from 19th-century colonists as well as exiles from communism; Bulgarians as a result of the population transfer with Turkey). The Russians and Bulgarians have mostly assimilated into Roman culture by now, though, and speak the Greek language.
The Roman Emperors are directly descended from the Russian Tsars, but now speak the Greek language rather than the Russian language.