I'd layer that chicken wire with the welded wire all the way up and all the way around. It's cheaper to use the right materials, the first time around rather than having to constantly reinforce things. And welded wire is a much stronger and safe choice over chicken wire. It doesn't cost much more either. If the birds will be spending nights in the run, you will also need hardware cloth.
Xs2! I have welded wire fence all around and the top, coon have walked over my run. Do what you you can when you can and good luck. Just locked em up every night.