Try going to your public library and looking for books on how to build sheds. They have some that are pretty clear on how to build them. To convert a shed into a coop you need to add ventilation, roosts, nest boxes, and maybe a pop door. We can help you with those details.
In your climate I'd be tempted to look at a shed in the range of 8' x 8' or even bigger. It's a bit bigger than the absolute minimum some of us can get away with, but you are going to have some snow days when they can't get out. I also find it is less work for you managing them if you give them a bit of extra space. I serioulsy think bigger is better for them and for you, but bigger gets more expensive.
At that cost, you are not going to buy anything new. As others have said, look for used materials. Especially look for sheds people are giving away on Craigslist or something like that. Sometimes you can transport them and sometimes you can tear them down for the materials.