An interesting asside is that some historians say Charlemagne was "furious" when he was unexpectedlly crowned the "Roman Emperor" by the Pope. Charlemagne had built his empire without the help of the Church, but by accepting the crown it was as if he needed the Pope's authority to rule.
I think the issue (and the reason you are being asked this question) is that your teacher wants you to find out how different groups define a Roman Emperor.
The first post points out his possible knowledge of Latin and knowledge and use of Roman Law.
The second post defines a Roman Emperor as one who ruled the area known as Rome through centralized government and was part of an unbroken line of emperors.
The third post states that neither of these need to be true to define a Roman Emperor.
The key is there is no one definition of a Roman Emperor. We can only say that Charlemagne WANTED to be seen as a new Roman Emperor with all the history, tradition, power and image that that implies. Most would then quality Charlemagne as an "impostor" or "wannabe" Roman Emperor. The key is what factors do you or your class or your teacher use to define an Emperor of Rome and did Charlemagne meet those factors? Otherwise you have to make an intelligent decision for yourself about what criteria make up a Roman Emporer and does Charlemagne fit those criteria?
I personally think your teacher is asking you to look a little further into this and make some intelligent conclusions of your own. Good Luck!
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.
A Roman Emperor was one who ruled the area that was Rome and who did so through Roman institutions like a centralized government. A Roman Emperor got legitimacy from the fact that he came from an unbroken line of emperors (even though this legitimacy was stretched thin by all the political upheaval at various times). Charlemagne did not fit these. He came centuries after the "fall" of Rome and had no connection to those emperors. He was in a feudal system, not a centralized one. He was not a Roman emperor.
Charlemagne was attempting to recover some of the political legitimacy held by the Roman Emperors until the fifth century AD. He had brought much of Western Europe under his control by 800 AD, and when he supported Pope Leo III against attacks by Roman nobles, the Pope conferred the title of Roman Emperor on him, thus creating a Western Roman Empire in opposition to the Byzantines. As for whether he matched the description of a Roman emperor, what historians know is mixed. Charlemagne, according to Einhard, his biographer, could understand some Latin, and he attempted to bring about some legal reforms based on Roman law, as well as encouraging learning in Latin in monasteries and church schools. He also was often portrayed in a Roman style on coins. But he dressed as a Frank, spoke Frankish, and observed Frankish customs, both in battle and in diplomacy. The most obvious example of this is the division of his empire among his three sons upon his death, a Frankish custom that caused a great deal of political turmoil