The above answer is incorrect in several aspects. Charlemagne considered himself King of the Franks, and remained so until his death. His subjects considered him as King of the Franks also as well as they did his father, Pepin the Short, and his son Louis the Pious.
Charlemagne was crowned "Emperor of the Romans" by Pope Leo III on Christmas Day, 800 C.E. There is considerable debate about Leo's purpose in doing so, or even if Charlemagne knew he intended to do so.
Charlemagne spoke a Frankish tongue, and although he knew some Latin, he was illiterate. He spoke of his Empire as the Frankish Empire, and later generations have called it the Holy Roman Empire. He neither dressed nor conducted himself as a Roman Emperor, and considered the only true Roman Emperor to be the Emperor in Constantinople. So it did NOT depend upon whom one asked--everyone would tell you that Charlemagne was NOT a Roman Emperor by anyones definition.
This is a great question and the answer will depend on who you ask. In other words, there will be differences of opinion. Let me start off with a few words on who Charlemagne was.
Charlemagne (Charles the Great), was the king of the Franks from 768. He also took the title of the Emperor of the Romans from 800 until this death in 814. So, if we asked him whether he was the Roman Emperor, he would answer in the affirmative. If we were to ask his subjects, they, too, would answer in the affirmative.
However, if you were to ask the East Roman Empire, which was based in Constantinople (post Constantine), they would not acknowledge Charlemagne as the Emperor of Rome. As far as they were concerned, the western empire ceased to exist.
Modern scholarship by and large does not see Charlemagne as in the line of Roman Emperors. No list that I know of traces the Western Roman Emperors to Charlemagne. Scholars are generally accepting of the fact that the last Roman Emperor was Romulus Augustus (475-476).