An educated person certainly would have been aware of Rome, probably more than any other historical empire. A lot of this comes rather naturally from the fact that many of the school books, Cicero, Donatus, etc., and many of the classic works of Christian thought originated in the empire. Livy, Ovid, and Virgil were also still read. This only increased as time went on and the interest in classical texts grew through the "Renaissance." Besides that, Rome was commonly mentioned in histories and chronicles. For example, Honorius Augustodunensis, the author of a very popular encyclopedia entitled the Imago Mundi, included a brief survey of past empires in which Rome got by far the most space.

The period 1000-1600 is not the period usually associated with the Dark Ages. These are the High middle ages and Early Modern Period. Some historians refer to the entirety of the Middle Ages as the Dark Ages, but it is usually just used to refer to the Early Middle Ages.

And boy what a question, about the entirety of Europe over a period of 600 years. That is a lot of ground to cover.

Now the question of how aware is going to depend greatly on were in Europe and in which time period. The world in the year 1000 was almost as different from 1600 then 1600 is from now. And someone living in Rome or Italy will have a very different relation to the Roman Empire then somebody living in Finland.

But overal the answer is going to be, yes they knew it existed. And depending on where they lived, how learned they were and when they lived determined how much they knew.

First of all, there where in this time period at any point two states that called themselves the Roman Empire, with at their head the Ceasar. This being the Holy Roman Empire in the west, ruled over the the Kaiser and the Byzantine Empire in the East ruled over by the Basilius. Now the later is in many aspects the Roman Empire of old, it was called the Greek Empire by its contemporaries in catholic Europe (They had seized using Latin and its people spoke Greek), But by the people living in it it was called the Roman Empire and they saw themselves as Romans. So wherever you were in Europe, you were aware of there still being a state that called themselves the Roman Empire.

Secondly there is the architecture, the parts of Europe that were part of the Roman Empire are to this day scattered with buildings from Roman times. While the majority of these buildings are in Mediterranean Europe, they are also found in England, the Low countries and Germany. For example; Hadrian's Wall or the Porta Nigra. And that's not even talking about all the Roads and Bridges they left behind. People saw this architecture and knew it was build by the Romans, they were often still called the "Roman Road" or the "Roman Bridge" that knowledge didnt leave most parts of Europe.

And thirdly and maybe most important, the Church thought them. The Roman Empire is featured in the Gospel and plays a pretty big part the story of Jesus. Sure most people couldn't read the bible, but for them there were plays and other forms of media. There is even a feast day on the Roman Calender that commemorates a Centurion, Cornelius the first gentile to be baptised.

Now iam not going to present the Renaissance and the rise of Humanism as examples here. Yes, they were both characterised as the revival of Roman literature and art. But this was very much a revival for the educated and rich, people who would have been aware of the Roman Empire regardless of these movements.

So to recap:

Yes, most people would have known about the Roman Empire:

• There were successor states that still called themselves the Roman Empire.
• The Romans left many traces still visible to the people living in that time, often still associated with the Romans.
• The Roman Empires features quite extensively in the Bible, and would have been part of church teachings.

TL;DR: We don't say Dark Ages anymore, and even if we did they didn't start or end when you think they did. Medieval Europeans had most of the available sources about Rome that we have now. However, they understood and valued history, and thus Rome, differently than we do. Furthermore, it's actually impossible to answer this question for an entire continent across hundreds of years. We know that most educated people knew a lot about Rome, though.

First, you should reset what you think of as the "dark ages". Historians don't use that term anymore, as it tends to be pejorative and give people a very negative understanding of what the Middles Ages were like. Even when we did use that term, it certainly did not cover the period from 1000 to 1600. Rather, it covered the period from the end of the Roman empire to whatever point a given person or historian felt was less dark. This has ranged from roughly 500 to 900, 1100, or 1300, and rarely all the way to 1600. Bear in mind that by the 17th century (1600), Europeans were already well into the Reformation, Renaissance art was established and Baroque art was emerging, Columbus had discovered the Americas and colonization was in its early stages, Shakespeare's career was developing, Copernicus had theorized heliocentrism and Kepler and Galileo were taking it up, etc.

Instead, we tend to divide the Middle Ages into three rough periods, the Early Middle Ages, the Central Middles Ages, and the Later Middle Ages. These roughly correspond to 500 to 1000, 1000 to 1300, and 1300 to 1500.

Even here, though, there are some problems, which we tend to talk about under the umbrella term "periodization." The problem with periods is that they're completely arbitrary -- divisions that we've put backwards onto history rather than natural breaks. For example, the Reformation, which is largely considered to be part of (even the beginning of) the early modern period, began in 1517 (on October 31st, precisely, when Martin Luther posted his ninety-five theses). However, I've just told you that the "end" of the later Middle Ages was 1500. What do we do with that 17 year gap? And what does life look like after 1517? The reformation didn't transform Europe overnight; it was a gradual process that took over a century to fully develop. There's also a geographic element. 1500 in Florence, Stratford-upon-Avon, a tiny hamlet in Norway, Wittenberg, and Constantinople all looked different.

The uneasy solution to all of this is that historians still use period labels, because if we don't we have no agreed upon vocabulary to talk about the past. However, we don't accept the boundaries between the periods as hard edges, and we've done our best to move away from labels that carry too many positive and negative connotations (such as The Dark Ages or The Renaissance).

On to your question. As /u/wedgeomatic has laid out, Rome figured very prominently in medieval literature. Certainly any well educated person in the Middle Ages was extremely aware of it. What the average peasant thought, on the other hand, we can be less sure of.

Political figures attempted to use its legacy to bring themselves legitimacy (for example the Holy Roman Emperor). And Roman, and to a certain extent Greek, history played a strong role in contemporary medieval understanding of their own position within history. Consider the first lines of the Pearl Poet's Sir Gawain and the Green Knight, which traces the escape of Aeneas from the fall of Troy directly to the establishment of Britain and King Arthur via Aeneas's founding of Rome.

This passage, however, points to the very different ways that medieval Europeans thought about and used history. Our modern understanding of history tends to be more concerned with an objective investigation of what went on in the past. Historians are interested in how the past bears on the present, but we're less likely to make claims about the trajectory that history is on because we don't see any agreed upon endpoint (teleological history).

In the Middle Ages, however, a teleological and cosmological view of history was the predominant way of understanding it. History was important only insomuch as it taught its student about God and the divine hand on earth. Medieval Western Christians saw Christendom (ie their contemporary Europe) as the Kingdom of God on Earth and the way that God would eventually bring about the salvation of mankind in the end of the world. Rome was the height of the Pagan world, and was replaced by Christendom. Thus, medieval people understood Rome as a fallen precursor to their own more perfect but still fallen society. They were interested in identifying "Virtuous Pagans" like Virgil and honoring Julius Caesar as one of the nine worthies (ideally Chivalric figures from history). They did not study Classical history on its own merits, like historians do now.

Finally, in a way it's impossible to answer this question. What did people in the Middle Ages know about Rome? Most of what we know, actually, given that nearly all of our sources about Rome are medieval. However, what the average knowledge about Rome across time and space was is impossible to tell. Access to books about history, interest, available teachers, etc. would have differed so much across time, place, and individual that it's really impossible to say with certainty what the average person knew. It's also nearly impossible to say what "the peasantry" knew about anything, given that we have almost no sources by or about them.

Unfortunately, the main person I'm drawing on is Hans-Werner Goetz, who hasn't actually been translated much into English. Life in the Middle Ages is good, though, if you want to learn more about the Middle Ages in general.

There's a good discussion of the medieval uses of history in the beginning of Jeff Rider's God's Scribe: The Historiographical Art of Galbert of Bruges.

Also, here's a translation of Sir Gawain in case you want to read the first two paragraphs: http://www.yorku.ca/inpar/sggk_neilson.pdf