Alright! Since you mentioned Justinian, I'll be focusing my knowledge of the demographics on him and his era.

The people who "settled" the Roman East were almost entirely Greek or assimilated indigenous inhabitants of the areas. Asia Minor, a core region of the empire at the time had a coast that was predominantly Greek-speaking and an interior that spoke Greek mixed in with some indigenous influences (these influences would eventually completely die out). Obviously, Greece itself was Greek-speaking, and Moesia/modern-day Bulgaria we have good reason to believe spoke dialects that were Greek-influenced. Illyricum (think of it corresponding to modern-day Yugoslavia) spoke Latin-influenced languages, but the Slavic migrations beginning around this time would soon change that along with the entire demographic makeup of the Balkans.

Going further east, you had quite a bit of cultural diversity. In Syria, cities like Antioch were reasonably Hellenized, but the majority of the indigenous inhabitants still spoke Syriac languages. The plague during Justianian's reign shifted the demographic balance in favor of the Syriac-speakers. In Egypt, it was much the same, with cities like Alexandria having a majority of Greek speakers but the countryside and other cities being majority Coptic-speaking. A more important difference was that the Greek-speaking regions of the empire tended to follow Chalcedonian Christianity while the Eastern regions preferred the Monophysite "heresy."

Further west, with Justinian's recent conquests in Italy, you had a majority of people still speaking something that sounded like or was Vulgar Latin, unsurprisingly. Africa, the area around Tunisia today, probably also spoke some Latin-influenced language. Both these regions followed Chalcedonian Christianity.

The average early Byzantine (although Greek culture would be dominant in his empire) would call himself a Roman, or Romaioi. The term "Hellene" by that time had associations with the pagan classical past, which most Byzantines really didn't want to be associated with.

Generally speaking, Justinian's time was nearing the end of the period of the urban-centered civilization of Late Antiquity (it was pretty much shattered forever by the plague and the Arab conquests). Constantinople was probably the largest city, with an estimated population of around 300,000. Alexandria and Antioch were probably the next two largest. Thessaloniki, Smyrna, and Damascus were probably prominent among the "second-tier" cities.

The most prosperous region of the empire was Thrace and Western Anatolia, probably followed by or equaled to by Egypt. Syria was also significantly urbanized, though it suffered somewhat from Persian attacks like their sack of Antioch. The Balkans, with the exception of Greece, was lightly populated and heavily raided by Slavs, and eventually the Byzantine presence in the Balkans would collapse entirely. Italy's population had been hugely diminished by the reconquest and the plague, and it was really just an impoverished region unable to pay for its defense with some historical value. Africa by far was the most prosperous of the reconquered provinces: it produced a valuable agricultural surplus and was pretty easy to defend.

The big demographic change in the time occurred with the plague, probably the same as the Black Death that would strike late Middle Ages Europe with the same catastrophic effects. It greatly affected agriculturally productive regions like Egypt, with wet soils that apparently suited the carriers of the plague, and the densely populated cities like Constantinople unsurprisingly. I can't find the numbers in my current book, but if I recall correctly it probably killed something like 30% of the empire's populace. This was what really devastated the Empire and made it overextended: before, the great reconquests of Justinian could actually reasonably pay for themselves: afterwards, it was extremely difficult to maintain them.

EDIT: forgot my source!

A History of the Byzantine State & Society, Warren Treadgold