This question is a difficult one to respond to, since it's both difficult to quantify, and of course the early Middle Ages were right after the Classical era, so it was, essentially, yesterday to them. By the time you get to the Early Modern period, Shakespeare is still writing plays about classical figures for popular audiences (for example, Julius Caesar), clearly expecting even the groundlings to know who these figures were.

So, it strikes me that the answer to your question is probably "a lot." Heck, since the classical education wasn't really replaced by liberal arts education until the early 20th century, educated people in the Middle Ages/Early Modern Era almost definitely knew more than educated people in the 21st century -- but again, this is difficult to quantify, and so hard to defend.

Perhaps you could word your question in a more pointed way?

Like /u/ProfAwe5ome said, it's a difficult question to answer precisely, because it's so open-ended, but the answer is a lot. I will just give three examples of some common types of works available to scholars throughout the Middle Ages that would have helped them know about the Greek and Roman world. It was of great interest to them for a number of reasons, perhaps especially so that they could understand the details of the bible, but also to help advance other areas of knowledge such as law and medicine.

Histories: Many Roman historical texts were preserved and read, but perhaps more popular is something like Orosius's History against the Pagans, basically a history of everything that had happened in Rome from its founding from the point of view of someone proving that bad things happened when they were still pagan (because people complained that the sack of Rome in 410 was the result of conversions to Christianity and the abandonment of the old gods).

Scholarly works: Commentaries on Roman literary texts, such as Servius's comments on the Aeneid, explained a great many of the small details of the text that might have been unfamiliar to a later audience, much like the footnotes in a student edition of Shakespeare.

Encyclopedias: Isidore of Seville wrote a large work on the meanings and etymologies of Latin words that is really just a huge encyclopedia of classical knowledge.

Much could be added to this.