Was the emperor Constantine part of the Western Empire or the Eastern Empire?

Constantine was, in fact, among the first rulers of the Byzantine (East Roman) Empire, which is also known as the Byzantine Empire. The name "Byzantine" is a historical term and not one used in the time of Constantine, as during the time both "sides" thought of themselves as the Roman Empire. In 308 AD, the Roman Empire was split between the "Greek" East and "Latin" West--Constantine was appointed in the East. Constantinople was the capital of the Byzantine Empire, also sometimes referred to as the East Roman Empire. It was founded by Constantine in 330 AD to centralize the empire. He was also the first Roman Emperor to popularize Christianity and to declare the Christian faith as the official faith of the Empire. 

However, a few chief differences separated Constantine's East Roman Empire (the Byzantines) from the Western Roman Empire. The biggest is that the Byzantines spoke Greek and had Greek stylings. In fact, the Eastern Roman Empire was sometimes referred to as "Imperium Graeocorum" ("Empire of the Greeks") by those in the west, whereas the Byzantines referred to themselves as Romans.