Don't use chickenwire (IMHO). If this will be an open-topped and basically-not-very-secure run ANYhow, at least use something that animals can't easily rip through, like 2x4 welded wire (nothing with any larger opening than that, though). It would be very wise to add something with smaller holes over the bottom 2-3', too, to prevent reach-through 'incidents', but if your fence wire is strong like heavy gauge 2x4 mesh is, you can potentially use something weaker for the smaller-meshed material if you really, really want to.
Cheapest way to get wire fencing is to haunt farm auctions, or if you have a local junk or salvage yard that resells things, or put 'wanted - used wire mesh fencing' ads everywhere and keep an eye on Craigslist. Most of mine is secondhand. Secondhand chainlink works too, IF you know how to install it CORRECTLY (and have all the necessary pieces).
Good luck, have fun,
Pat