how do the characters speak

The quick answer would be that it was written in Middle English. The poem was only survived in manuscript. We don't really know who wrote Sir Gawain and The Green Knight but we do know that he used Cheshire dialect would not have seemed odd at the time . My info comes from a great handout to class that I used to give on this very topic. I'll give you the site for this in the source link below. It's comprehensive and deals with exactly your question.