Well, chickenwire (the hexagonal mesh stuff with relatively thin wires) will keep your chickens in but will not keep any serious predators *out* (raccoons, foxes, dogs, coyotes).
While you are going to all the trouble of making the run in the first place, I would suggest saving yourself the labor of doing the whole darn thing over again after you lose some or all of your chickens to a predator -- use 2x4" or smaller mesh right from the start.
(If you use anything larger than 1x1" mesh, it would be wise to run hardware cloth or at least chickenwire around the bottom 2-3' of the fence as extra insurance against things reaching thru... although if your chickens will always be locked up in the coop by dusk this is less of an issue). (Also, some kind of skirt or buried section is wise defense against digging predators).
There are three common ways to attach wire to a run. In approximate descending order of strength:
1) Screw wood battens (1x3+, or even 2x4s) down so that you are screwing thru the mesh and into the wood frame. Or
2) Screws with fender washers (this only works well for 1x2 or smaller mesh). Or
3) Fence staples -- not the kind you use a staplegun for, but the kind you hammer in. They are very very aggravating to work with, btw.
Hope this helps some,
Pat