Chainlink works fine for a run, you just block the gaps off with pressure-treated boards or some such thing. Attach smaller mesh wire around the inside of the walls for the bottom 2-3' or so to prevent reach-through, and unless it's on a slab attach an apron of several-foot-wide 2x4 welded wire mesh lying on the ground; and you're good to go.
However I would worry that, from appearances and price, that may be very flimsy chainlink. There are flimsy chainlink grades on the market that dogs can rip apart, and you don't want that for a chicken run IMO.
If you want a chainlink dog run for your chickens, either pony up for a heavier duty one, or hang around til you can find a secondhand good-quality one for sale.
Good luck, ahve fun,
Pat