I dont have chickens yet, but hope to in the future! Here's some ideas Ill try once its time.
Look into hoophouse setups using galvanized fence top-rail...they are super cheap and easy, bend it yourself. Lightweight too, awesome if you plan on moving it (look up mobile chicken tractor). You can use plastic, wood, fencing, metal panels etc. to finish it how you like.
Ive been watching Edible Acres on youtube and they do a small scale chicken run/compost system utilizing a hoophouse. They also do a deep compost system inside the coop...lots of hay!
For the actual coop structure, you could probably get away with using pallets. Plenty for free around here and usually the wood is hard and rot resistant, not like standard pine.
Depends on how nice you want it to be. Scrap wood and chicken wire is where I'd start. If you're in a cold climate consider some tarp or fabric panels to block the wind. Make sure all the little gaps are filled in.