If you get a fair amount of snow, having a covered run will allow your chickens to enjoy the outside even if they don't like snow (mine don't). You will also stop worrying about them when it rains. Finally, it's really great for predator proofing.
I'm thinking that if your run is not framed in (just stretched wire and posts) you are probably not going to be able to put on a roof that will stand up to snowload. Without a photo, it's hard to say. In general, flat roofs and snow aren't usually a good combination.
If possible, frame your roof with 24 inch centers and slope the roof. Mahonri is right - you'll want to make sure you have water control for runoff, whether it's downhill or controlled by a gutter. I put recycled tin siding (free!) on my roof and screwed it down to the framing then silicone'd the holes to reduce leaking. Works great.