Anywhere from $20,000 to absolutely free.
Really, it depends THAT much.
If you are a good scrounger, living in an area that lends itself to scrounging (esp. of run fence wire), willing to wait until you have scrounged all the stuff you need, your costs can be extremely low. There are even BYCers who have done it for free although most normal people would at least buy new fasteners (nails, screws)
If you are more of the Martha Stewart, "I saw it in a magazine and I want what I want and I shall have one of my people build it for me" type, then it can get real expensive real fast.
Or anything in between.
FWIW the bit that's hardest to find for free, in many areas especially urban areas, is wire mesh for the run. SUITABLE wire mesh, i.e. not chickenwire and not too flimsy and holes not too big and not in too terrible shape. For 4 chickens, an adequate (not overly spacious) run would be like 4x10, ish, which if one of the 4' ends is your coop itself, would require only a 50' roll of 36" 1x1 welded wire mesh. You can price that locally. (That 50' roll would give you either a 4' high run, awkward to work in, PLUS a 2' digproof apron; or a 6' high walk-in run with no apron so you would have to get additional wire or scrounge some pavers or *large* flat concrete rubble to digproof it, unless the run is on or surrounded by concrete slab. In both cases, a *top* for the run would require extra materials)
As a ballpark, though, I'd say if you want to build it yourself and go for economical-though-tidy-looking, and you pay full retail for most of your materials, you are probably looking at between $500-200 for a minimal 4-chicken coop, more if you go bigger (which you may ought to if you're in a very cold-winter climate)
Good luck, have fun,
Pat