Sunny or shady?

Clematis comes to mind for being the most well behaved (won't try to take over the world)

There are VERY many kinds, all bloom at different times and are repeat bloomers too. Getting a variety of them will give you flowers throughout the growing season including fall

Our neighbor has a wisteria vine that covers all 3 1/2 sides of his chain link fence. He keeps it pruned (after flowering) so that it stays pretty neat. It is green all summer and of course is covered in flowers in the spring. Another fence nearby is covered with our native type of passion vine (maypop) and it grows REAL fast. The only problem with it, is that is has to be sprayed or the Gulf Frittilary caterpillers eat it to stems in the summer.
Betsy

morning glories, or i love honeysuckle, they smell absoluetely awesome!!! if ya want hummingbirds to come over grow trumpet vine in some of it, at least with the morning glories you can plant all kinds of different ones!! LOL , and if ya planted a few honeysuckle there, it'd have a little smell. we are fixing to put up a fence because we have a pool we are putting up, so ya have to have a fence, so we are going to plant some of the different morning glories and passion vine, and the honeysuckle, i sooo want the smell!! LOL ohhh ya and pandorea vine. :')) good luck!! Medo and yes!! i'm a little crazy and nuts!! LOL
Smiley Face

Ooh, I would definitely put a variety of vines around, different sections of fence could have different looks! And it could all ultimately blend together.

Clematis is gorgeous, and perennial. Morning glories are beautiful and vigorous, and although an annual it reseeds well. Sweet peas, also annual, are lovely and smell wonderful. Passion vine is very vigorous, perennial, and has interesting, unusual flowers. Chilean glory vine is nice too, with tiny leaves and striking flowers.

You could play around with leaf size and shape, bloom time, etc. in your choices of what to put near what. I think it could be a glorious sight if you have a lot of chain link to fill.