Hi there!

So right off the bat, Constantinople was established in a region that spoke predominantly in Greek. Even before Constantine, many aristocrats, emperors included, had some proficiency in the Greek language, so this linguistic change wasn't out of the blue. It had been the academic language of the cultured upper class since the time of the Republic (similar to how modern college graduates show off their French or Latin). Many emperors before and after Constantine I spoke Greek in that way, knowing enough to display their good education, but not enough to be fluent. As time and generations passed, the imperial family began to adapt to their environment, becoming true Easterners, not Westerners living in the East. By the age of Justinian I (the mid-6th century CE) Greek culture had become naturally dominant among the imperial family after over two centuries of immersion. Greek speaking spouses, advisors, usurpers, and clergymen were in constant flux in the imperial class of the East.

That brings us to the emperor Heraclius. Up to this point, Latin had fallen out of use as the common vernacular of the Eastern nobility, but remained as a formal language for the military and administrative bureaucracy. In the early 7th century CE, Heraclius (a native to the region) overthrew Phocas, a royal usurper who had deposed the emperor Maurice. In 620 CE, Heraclius made a play to increase his legitimacy and consolidate his empire: he officially renounced his ties to the memory of the Western Roman Empire, and made Greek Koine the official language of Constantinople. He even changed his title from the Latin "imperator" to the Greek "basileus". What had been an organic shift was now official and set in stone.

Tl;dr: The transition from Latin to Greek was a natural result of time and cultural blending, but Heraclius made the switch official in 620 to break completely with Western tradition.

Sources:

Giannouli, A. (2014). Education and Literary Language in Byzantium. Studies in Byzantine History and Civilization The Language of Byzantine Learned Literature, 52-71. doi:10.1484/m.sbhc-eb.1.102124

Gregory, T. E. (2011). A History of Byzantium (2nd ed.). Malden, MA: Wiley-Blackwell.