Welcome to BYC. You are headed down the right path to protect your coop / run. You can use 1/4" hardware wire, or if it is cheaper, you can use 1/2" hardware wire. If you have it 24" high, that will prevent a large predator from reaching in and grabbing a chicken who is near the fence. But chain link fence is not a great fencing option. The openings are large enough to let predators through. My dogs cornered an opossum in their run, which is chain link, and it started to escape through the chain link fence. I called off the dogs and thought I was going to have to go rescue the critter because there was no way he was going to fit through. Low and behold, he made it through. I would have bet $100 he was going to get stuck! So while hardware wire around the bottom will provide protection, the chain link above will be a weak spot for things like opossums, weasels etc. If you can, use hardware wire all the way from top to bottom. It will cost more yes, but will provide much better protection. You can use zip ties to attach the hardware wire. Put it on the inside of the chain link fence. A predator will pull on the wire, so if the hardware wire is on the inside, the chain link will keep it in place. If it is on the outside, it can separate from the chain link.
On my run, I have a double layer of fencing. The framework is 4x4 posts, and I have 2"x4" field fence on the outside, and 1/2" hardware wire on the inside. The two fences are separated by 3 inches, and I have never had any losses when the chickens and ducks are inside the pen. I have had a bobcat climbing on the outer fence, and it was able to bend it up a bit, but it could not touch the inner fence. The birds were safely locked up in the coop for the night, so even if it got into the pen, it would then have to break into the coop.
Here's a picture of my coop. The run has the double fencing.