If you'd studied Middle English before, it probably wouldn't be insanely difficult to figure out how to speak and understand it, especially if you have some knowledge of French (not even Norman French, necessarily) and are skilled at learning languages.
The main thing to keep in mind is the fact that Chaucer's Middle English was 1. London dialect, and 2. written in verse (for the Canterbury Tales, at least).
If your character is somewhere else, like off in the Northwest of England or something, then the dialect will be much harder. It's the difference between reading Chaucer and Gawain and the Green Knight. They were written around the same time, but are vastly different. I can read Chaucer pretty well, but Gawain...well, let's just say I didn't get far before switching to Modern English.
As for the verse, people didn't speak in rhyme while Chaucer is in rhyme. Sometimes, the natural pronuciation of a word wouldn't be exactly as you'd say it in the poem. Stresses were placed differently in poetry, and syllables might be drawn out. If you take a look at something like Chaucer's Treatise on the Astrolabe you'll see what regular writing, and probably speech, would have been like.
But, yeah, in general...it sounds more French/German-ish (depending on the word) than Modern English.To me, at least. Though, at the same time, I only know about later 14th century Middle English...It's quite possible that 13th century Middle English would have been closer to Old English (and, thus, probably closer to that of Gawain instead of Chaucer's works) so it would sound much more German and would have tons of words that you probably wouldn't know unless you'd learned them before.