One of the best aspects of Roman Empire Charlemagne continued the tradition of was culture and learning. As Rome was an empire of culture and learning, Charlemagne sought to elevate his kingdom to an empire of culture and learning. For instance, he encouraged literacy and introduced the handwriting script called Carolingian minuscule that consisted of both upper and lower case letters. It replaced the all-upper case scripts previously by scribes and monks.
I would want to argue that Charlemagne was rather shrewd in the way that he carefully adopted aspects of Roman culture that benefited him and his position whilst clearly and defiantly remaining a Frank. This was something that enabled him to have the best of both worlds, in one sense, as he promoted his Frankish identity whilst also learning and benefiting from Roman experience.
I would say that Charlemagne was a Roman. We need to realize that the Roman empire changed so much. Transformation continued to take place. For example, Constantine introduced Christianity and moved the capital to Constantinople. The later emperors had greater exchange with "barbarian" tribes and so on. In light of this transformational view, Charlemagne was another step in that transformation. The fact that he did take the title of Roman Emperor says a lot.
As the previous posters have noted, Charlamagne was by no means a "Roman" in the sense that, say, Julius Caesar was. However, the value he accorded to Rome as a source of cultural inspiration and as a model of political unity was important in helping to perpetuate the influence of Roman culture in later generations.
Rome never really "fell" as historians have claimed for several centuries; it did evolve into something quite different than what it had started in antiquity, from a loose alliance, to a republic, to an empire, and finally to just a cultural force. Of course, the Empire collapsed, but the influence of Rome was felt for centuries long after the traditional date of 476 when the Western Empire became more and more politically and economically impotent.
Charlemagne (742-814) ruled quite a bit later than the Empire; he was, however, a strong enough ruler to keep most of the former Western Empire cohesive, at least for a few generations after him. Was he a "Roman Emperor?" Only in name, but he did use the cultural artifacts of Rome to keep his portion of the former Empire intact. Ironically, the concept of "Roman Emperor" continued for a millennia, until Napoleon finally did away with the "Holy Roman Empire" and its office of "Emperor" in 1804! See the link for an interesting list:
I tend to agree with the first post here. However, if you want evidence the other way, you can point to the fact that Charlemagne was Christian, which is a direct effect of the Roman influence on Europe. The fact that he conceived of creating a "Holy Roman Empire" shows this as well. So, in an important way, Charlemagne was Roman because he worshipped a god who came out of the Roman Empire and he thought of empire the way a Roman might have.
Also, I suppose I should note that he did speak Latin and understand Greek.
As referenced in the other question on the topic, Charlemagne adopted Roman imagery in the form of coins, and he did accept the title of Roman emperor from the pope. These were political decisions aimed at increasing his own legitimacy. But ultimately Charlemagne dressed like a Frank, spoke Frankish and ruled and fought wars according to Frankish custom. His laws and epistolary decrees were heavily rooted in Frankish customs like the wergild, trial by ordeal, and Frankish marriage rituals. He was a Frank, not a Roman.