Chainlink is fine IF IT IS INSTALLED CORRECTLY and is a heavy gauge to begin with (the cheaper chainlink sold these days is not strong enough to keep dogs from going thru even if it *is* installed correctly). It is not at all easy or inexpensive or fast to install chainlink correctly though, especially if you have not done it before. Your posts need to be well-set and absolutely even and level; you need a top rail of the same pipe you're using for the posts (you CAN use wooden posts for chainlink but it's a whole lot more annoying to do) and a heavy bottom tension wire (or rail or board); you need to strain the chainlink quite tightly but not so tightly you damage/distort it; and it needs to be correctly attached at all points.
If you've done chainlink before, or if you have ALL THE PIECES from a CORRECT previous installation and are willing to be very careful about following directions, you can certainly do it.
Otherwise however you should not regard it as a predatorproof fence; if it is incorrectly installed, for instance too wibbly, animals can pretty easily just pry a hole in it and go thru.
If budget is a concern, and chainlink cannot be done properly, I would suggest maybe doing a good job on a smallish run with 2x4" wire mesh, configured so that you can enlarge the run later on as funds become available.
An apron of chickenwire and/or LARGE pavers is a reasonable digging barrier -- I mean, chickenwire is not ideal for the purpose but is not bad, especially compared to the cost and aggravation of burying the base of your run fencing (which is exceptionally difficult to do with *chainlink* unless you're going to do a crappy incorrect job of installing the chainlink)
Good luck, have fun,
Pat