A Roman emperor was not defined as someone who ruled Rome. Nor did he get legitimacy by inheritance through an unbroken line -- that would disqualify the majority of actual Roman emperors (including all of the Antonine and Severan ones).
By the time of the end of the Republic, Rome as already a vast empire, stretching from Italy through Greece, Asia Minor, Spain, and Africa. Although the city of Rome remained important in the west, Constantinople, the eastern capital, was equally important after Constantine founded it.
The issue of Constantine's legitimacy as a Roman emperor had to do with the relationship between, as the first poster pointed out, the Bishop of Rome (and his claims to be "Pope") and the locus of imperial power in the east.