The Byzantine Empire was originally an offshoot of the Roman Empire. The Roman Empire was built as a world empire, made up of a number of client states. However, the official language of Rome was Latin.
This continued as the language of the Byzantine Empire starting from when it was founded in 310 A.D. However, as time passed, the Byzantines realized they had to move away from their Latin roots and pick up the Greek language of the locals since they were no longer based in Rome, but in Constantinople and the Hellenic world. So, in 610 A.D., Heraclius introduced Greek as the official language of Constantinople.
As a related note, the divide between the Latins and the Greeks in the political realm also had an impact on the religious realm, resulting in the Eastern Orthodox Church being divided from the Roman Catholic Church. It is interesting to note that as a legacy of the erstwhile Byzantine Empire, the Patriarch of Constantinople today remains the “first among equals” in the Orthodox Church.