My experience with the thin guage welded wire mesh is that the welds break apart. If this is the same material you plan on using, I'm not sure it will hold up well to a dog who is bored and wants out.
But if this is what you want, then I guess the usual way is to run top and bottom 2 x4 rails and nail the wire mesh to it. It's basically a fence, after all. The posts could be 4 x 4 or possibly 2x6, except that some are going to say not to use pressure treated material that your dog can chew.
You probably come out ahead just to buy it, especially if you can find it used or find two 10 x 10's you can tie together. Chain link fenced panels can be easily joined with bolt on connectors. By the time you buy parts and assemble them, I think you probably will come out better buying it premade or used unless you have a wholesale source for the parts. Chain link might not be the most pretty, but it is very practical for this application.