The most important thing to note here is that Medieval Greek can be seen both as Greek written by people living in what we today call the Medieval Era, and which generally speaking (borderline cases apply) we call today Byzantine Era when it relates to the Roman Empire, and it can also mean the Greek being spoken at that same time.
The written Greek follows, by far, the conventions of Antiquity, with Koinê and Atticism as the leading conventions. This means that if you know Classical Greek you can read what Byzantine writers wrote, with only a small number of words being added, some others getting a new meaning, and some very fine tuning of the syntax (but then again the jump won’t be greater than, say, going from Thucydides to Polybius).
The spoken Greek, which obviously ended up as the Demotic or Popular language of more recent times (which, when paired with the XIX century Katharevousa, spawned Modern Greek), does not survive much, that I am aware. I may be quite wrong about this, but I wouldn’t expect us to have more than a few songs written in “pure” Demotic Greek (much in the same way that we barely have any “Proto-Italian” of the 10th century).
Then there a number of texts which are written in the scholarly language, but which adapt a higher number of variations from the spoken language, creating a sort of mixed language which was probably closer to what people would have spoken when they were trying to actively speak “Attic”, since it must be tough to maintain a Demosthenian style all afternoon, and words from your daily speech would have certainly creeped in. The most famous examples of this, again that I know of, are the Spiritual Meadow and the Digenis Akritas. Regardless of all that, they are still by and large readable by someone who knows Classical Greek, but the differences will be legion (to start with, a consumation of the abandonment of the quantitative meter of Antiquity, which was kept on life support by everyone else).
If you are looking for texts to read, apart from just suggesting a History of Byzantine Literature, maybe you could either take a look the Dumbarton Oaks Medieval Library, and check the Greek texts that they published, or else read Nigel Wilson’s Anthony of Byzantine Greek (I believe this is the title), a great book that covers both the continuity and the rupture of Greek language and style during the period.
Hope this helped.