Hi Luke.
Thanks for your interest.
Well the Roman Empire was a state more or less similar to the USA. People with different ethnic backgrounds eventually (after 2-3 generations) become American citizens. Likewise, all the people of the Roman Empire became eventually Roman citizens, even though they were from different ethnic backgrounds. There were two languages spoken in the Empire: Latin and Greek. Both were acceptable. Latin was spoken by just a minority (even before Christ, that is, even before the first century). Remember that the three out of the four Gospels and all the Epistles were written in Greek. The Epistle of Saint Paul to the people of Rome was written in Greek. That means that the majority of the people, even in the city of Rome were Greek-speaking.
Both Greek-speaking and Latin-speaking citizens were all Roman citizens and consider themselfs as brothers and co-patriots and people of the same nation. However, he main characteristic, the main feature of the Empire (especially after the 3rd century) was Orthodoxy. The main axis of the Empire was Orthodox Christianity. The Roman citizens (either Greek- or Latin-speaking) were first of all and above all Christians. Romans were proud to be Christians and not pagans/barbarians. They were proud to be the light in the darkness of the other civilizations. The Empire was the Christian State on Earth. That was their belief and their major notion.
When the Franks and other barbarians conquered the Western part of the Empire and in order to control the enslaved Romans and cut them from the free part of the Empire they started calling the free Romans as Greeks (Imperatorum Grecorum). They were saying to the enslaved Romans that the free Romans (in Eastern part) were not their brothers but something different, another nation (Greeks). And on top of that, that the Greeks were heretics and not real Christians and that themselfs (the illiteral barbarian Franks and Germans) were the real Cristians and the discentands of the Empire (Holy Roman Empire of the Germanic Nation or something like that).
Anyway, Constantinople/Nova Roma, the capital of the Empire was conquered by the Ottomans in 1453. But, even now, after six centuries the Turks still call the 2,000 Greeks left in Constantinople (last descendans of the Empire in the city) as Rum Ortodoks (that is, Roman Orthodoxs). See the title of the Monastery of Holy Mary in Balikli, Constantinople in Turkish here: http://romiosini.wordpress.com/2009/09/28/παναγία-μοναστήρι-ζωοδόχου-πηγής-βαλ/.
Finally, I must say that the majority of the Turks (especially of the Western part of Turkey) are enslaved Romans forcedly (and some of them willingly) converted to Islam.
Luke, I hope that all the above were usefull to you.
Best regards.
John