We built our coop out of recycled materials. The coop ended up being almost FREE. The run - - we had to pay for the wire.
We contacted local fence installers. We found one that gives away the old wood fences that they have removed when they installed new wood fences for customers. We used that to put together our coop. Some of the fence panels needed a 2 x 4 to reinforce the old ones.
We live in Florida, so normally the cold is not a problem. This winter, we stapled cardboard to the inside of the fence panels ( that way neighbors wouldn't see it) and it served as a wind break and insulation. When it warms up, we will take off the cardboard and the chicks have the spaces between the board for ventilation.
We used tin for the roof of the coop.
All together, we spent less than the budgeted $250 and have a 5 X 8 coop with a 8 x 12 run.
Lots of space for way more than 3 - 6 chickens. So, you could down size on the wire for the run and be just fine with the budget.
SO, Check with you local fence installers.