This is a very broad question, so I will try to answer briefly your minor questions and return with any follow ups you have.

so what happened to Greece during the Medieval period? Was it a part of a larger empire?

The period you are asking for is a thousand year long.

If we accept the medieval period is from 476AD (the fall of Rome) to 1453AD (fall of Constantinople), then mainland Greece for the most part was a province of the Byzantine Empire. Greek region was invaded several times through the centuries by Avars, Slavs and Bulgarians and was occupied briefly by the crusaders of the Fourth Crusade and finally by Turks in late 14th century.

After the Western Roman Empire fell, what happened to Greece?

Greece remained a province of the Eastern Roman Empire.

Who was living there?

Mostly Greeks is a short answer. But people of that time wouldn’t identify themselves as Greeks (Hellenes) and to fully understand this we have to examine the differences in identities people of the Greek region developed from the Classical period, through the Hellenistic to the Roman years and finally Christianity. People called themselves Romans, as the term “Greek” referred to idolatry, the polytheistic religion of the past. That said, apart from Greeks, groups of nomadic people settled in Greece during the Medieval times, such as Slavs, Avars and Turks in a smaller degree.

Were people still aware of Greece's contributions to philosophy and art and such?

Hard to say. They certainly knew that Parthenon in Athens for example was an ancient temple but not to the point to recognize the importance of that temple as a monument of the Athenian magnitude in the classical years proven by the fact that it was converted into a Christian church.