The Byzantines were always relatively interested in antiquity, and especially in Rome. They never "forgot" that they were, however theoretically/distantly, part of the Roman empire. I can't say for certain how much Constantine XI was interested in this, since he probably had more pressing concerns than ancient history. But the Palaiologoi do seem to have been very interested in their connections to pre-Fourth Crusade Byzantium, and in turn to Late Antique and even ancient Rome.

For example: the Souda is a relatively well-known Byzantine reference book, a kind of encyclopedia containing mostly interesting grammatical and historical trivia, originally compiled around the year 1000. (You can check out a pretty handy translation online for free; see the entries for e.g. Augustus, Caesar, the Roman Empire, etc. for what the Byzantines might have been interested in on these subjects.) Now, the Souda is written in Greek, and the Byzantines didn't know much (if any) Latin, but aside from the grammatical component of the entries (mostly, for example, words that appear only once in Homer/Sophocles/etc.) a great deal of the Souda's entries are on Roman history. There's certainly some Greek stuff, but not the kind of classical-interest stuff you'd get in a modern ancient history course: they've got a pretty substantial entry on Alexander the Great, but no single entry on "Athens" or "Sparta." When they quote historians like Polybius or Plutarch, it's mostly for what they've got to say about Rome, and almost certainly about imperial Rome (since the Byzantines too recognized the importance of having an emperor).

The tenth century was a remarkable period for this sort of thing, so maybe the Souda is just a fluke. But the Palaiologoi, at least two hundred years after the Souda was compiled, made tons of copies of it: Pinakes lists some 53 manuscripts copied between 1204 and 1500. Some of these are pretty fine copies, if not terribly embellished, but even a plain hand-written copy of an encyclopedia represents a tremendous amount of time and work. The manuscript I linked here is dated to 1402, so it's probably not originally meant for a western audience (as some of these, especially after 1453, may have been); this also means it was written when the empire was not necessarily in the best position, diplomatically or financially. Yet even while they're surrounded on all sides they're still copying books about the Roman past. Not all of the information they have about this past is accurate -- but at this point, the same is certainly true of the West.

I'm not terribly knowledgeable about Palaiologan classical reception, but most of the scholarly reading on the Souda that I'm familiar with is a.) in Italian and/or b.) probably difficult to get to outside of an academic library. If you can get to it, I recommend Barry Baldwin, "Aspects of the Souda," Byzantion: Revue Internationale des Études Byzantines 76 (2006): 11-31. The S(o)uda On Line site also contains some general information about the work, the most relevant of which you can find here (under "The Raw Material").

I can't speak as well to the last Byzantine emperors but its early and middle leadership certainly were aware of their past connections and history to the earlier "Roman empire." This awareness, if nowhere else, is apparent in the Byzantines' own term for self-identification: Roman (the identification of "Byzantine" is attributed to Hieronymous Wolf in the 16th century).

Justinian I (527-565 CE) was driven to recover, through the Byzantine Empire, the lost glory of the former Italian Roman Empire. Despite the proclaimed cause of his Gothic campaign (the assassination of Queen Amalasuntha), the 535-554 CE conflict -- in addition to his Vandalic campaign in North Africa -- was focused around reclaiming lost Roman territory. Unfortunately, while Justinian was successful in that expansion, in typical Byzantine fashion, those gains were quickly lost in a relatively short span within the ensuing generations.

Heraclius I (610-641 CE) was a capable military general-turned-emperor from Carthage. He was certainly aware of his city's past history and affiliation with the previous Roman empire.

Later, several major religious events would bring that past mutual history to the forefront and its breaking point. The Monothelistic and Iconoclastic Controversies (First and Second) of the seventh, eighth, and ninth centuries CE, surrounded theological ideologies on the nature and will of Christ, and the legitimacy of icons. These matters resulted in Byzantine conflicts with the Roman papacy and Italian peoples. Those conflicts brought that past relationship to the forefront, but stressed that lingering goodwill toward a social, political, and religious break.

By the time we get to later emperors and dynasties, such as the Macedonian from 867-1025/1056 CE (I recognize the end of the M. Renaissance as Basil II), Byzantine identity is something unique. Although they still consider themselves Roman, the Byzantines are now the product of over 700 years of cultural exchange and conflict: their state reflects the integration and opposition to multiple foreign peoples, including the Rus, Bulgarians, Greeks, Goths, Franks, Persians, Jews, and Arabs. Emperors have come and gone, some by peaceful means and others by assassination or civil war. Consistency in the empire, I've seen it argued, is held by a break between perceived reality and actual reality: the elevated idea of the imperial court, versus the dirty reality of its complicated leadership (See the 10th century Book of Ceremonies).

This shifting perspective is even more apparent during in-fighting among Byzantine leadership post-Manzikert (1071 CE) when several aristocratic families actively aligned with Turkic forces for their own gain. The Crusades serve as another example (1095-1291 CE), in that the Byzantines were not always aligned with the Papacy and Western Europe's crusaders.

Alternatively, you should also consider the evolving state of Byzantine education and leadership over the centuries. Without going into too much detail, the seventh and eighth centuries CE can be considered rather fractious and tumultuous times. They are characterized by a shift in political leadership, away from educated scholars and toward former military commanders. This is as much a result of internal policies as the state's then-current military opposition and frequent engagement with Arab and Bulgarian forces. Internally, this period is recognized by a general decline in the empire's educational standards; public works within this time tend to drop off, to the detriment of the empire's previous standards and its citizenry. This decline is later corrected during the Macedonian Renaissance (867-1025 CE).

In short, the Byzantines (as we know them today) were certainly aware of their past history, as past glory and power were frequent points of reference in the empire's early years. However, the importance and value of that past history was subject to change over the 1123 years of the empire's run (starting from Constantinople's dedication in 330-1453 CE). Over the evolving course of its length, the empire was exposed to new ideas and peoples, and was subject to shifting demographics and political leadership. These changes repeatedly brought about policies and internal divisions which affected foreign policy, and often pitted it against past allies and affiliations.