There were some with considerable knowledge of Greek writing, and this knowledge did increase with the Scholastic tradition (but that wasn't geared towards Homeric works). All in all, the first full translation of the Homeric works came to Western Europe from Constantinople, via Italy, in the 15th century. Previously, a shortened Latin version -- the Ilias Latina -- was widely known in the West. It was a standard part of any decent education. This version, probably created in the first century AD, was barely above a thousand lines.
In Western Europe, the general "understanding" throughout the Mediaeval period was that more than this compressed version of the Homeric account wasn't needed; Homeros was regarded as a story-teller, whose account of events was unreliable. The West instead put stock in a couple of noted forgeries(!) from Late Antiquity. First, there's the work Daretis Phrygii de excidio Trojae historia. This was supposedly the eye-witness account of Dares Phrygius, who Homeros tells us was present at Troy, a recorded in Latin by Cornelius Nepos. But it's actually a fifth century fake. Then, there's Dictys Cretensis Ephemeris belli Trojani. Supposedly a sort of diary of another warrior who was present, Dictys Cretensis, as translated into Latin. But actually also a fake, from the fourth century. (In this case, a Greek 'original' did exist, but was of a late origin itself.) The only other known "source" that was used in the West was the Excidium Troie, which has been famously discovered among the manuscripts collected by Richard Rawlinson. We don't know the origins for that one, but it's just as incorrect as the other two manuscripts -- just in different ways.
Based on these dubious "sources", the Mediaeval "Matter of Troy" gradually coalesced: a collection of highly popular chivalric romances and such things. Thus, we get Le Roman de Troie by Benoît de Sainte-Maure (written c. 1155 - 1160), De bello Troiano by Joseph of Exeter (written c. 1183), the Middle English narrative poem The Seege of Troy (sometimes The Batayle of Troy), Konrad von Würzburg's Trojanische Krieg (13th century), Guido delle Colonne's Historia destructionis Troiae (1287) and Geoffrey Chaucer's Troilus and Criseyde (completed in the mid-1380s). The spurious souurces used for these works explain why they indeed do not imply familiarity with Homeros!
Nevertheless, we can say that the tale of the Trojan War was very widely known in the Middle Ages, albeit in a garbled form. It's everywhere. There's even the Trójumanna saga: a 13th century Icelandic translation (and distinctly Nordic re-interpretation!) of the work ascribed to Dares Phrygius. The story, distorted as it was, certainly held a prominent place in the imagination. To the point that virtually every royal house in Europe claimed to be descended from at least one of the heroes from the Trojan War.