You have lots of choices, especially for a sunny location. Most types of clematis, honeysuckle (not lonicera japonica though, it's invasive), or grapevine come to mind. We had a grapevine growing on our chain link fence when we moved into the house. The thing was massive and grew very quickly. Some people love it, we don't, so we cut it down. I didn't get around to cleaning up all the lower branches I lopped off, and within a few days the branches - we're talking about sections about two feet long and no longer attached to the base/root system of the plant - started sending down roots and sending up new vines!

What might work best depends entirely where in the world you are located, so give us a zone or better yet, a somewhat specific geographical location. This also will have a bearing on what might or might not be invasive in your area. And do you want the vine to hide or obscure the fence completely (e.g. something evergreen) or just something decorative to provide fruit or flowers? How much are you into care or pruning?

There are quite a number of wonderful perennial vines that grow in NH. (I am assuming that the NH in your user name indicates location, but if not these may not be good suggestions.) Some of the non-invasive honeysuckles include Trumpet honeysuckle (Lonicera sempervirens), European honeysuckle (L. periclymenum) and a couple of hybrids, Goldflame and Mandarin.

Lonicera sempervirens 'Major Wheeler' last week

There's a wide range of clematis available in sizes ranging from just a few feet to 15 feet or more, and they come in shades of purple, pink, reddish, and white.

Clematis Mikelite late June

Clematis Gravetye Beauty early July

Clematis Prince Charles early July

Clematis Venosa Violacea early July

Clematis HF Young early June

Clematis Ville de Lyon mid June

The native pipevine (Aristolochia macrophylla) is mostly a foliage plant, with large heart-shaped leaves and pipe-shaped flowers tucked into the foliage. It provides food for one of the types of swallowtail butterflies. Another vine that can be grown for foliage is hops, Humulus lupulus, which is another really large plant. Besides the various green forms that provide the hops for brewing and their ornamental value, there is a gold-leafed form. There's a great photo of mature hops vines in this thread.

Climbing Hydrangea will grow in part to full shade and has white flowers in June. It gets to 20' or more if untrimmed and needs a surface to climb such as a masonry wall or tree trunk so I don't think would work for your chain link.

Virginia creeper (Parthenocissus quinquefolia) isn't great for a small garden as it is a large, vigorous plant, but is good for a woodland edge where it produces fruit for birds and has crimson fall foliage. It can be grown successfully on chain link fencing also.

If you are interested in edible fruit, in addition to grapes, hardy kiwi vines produce clusters of grape-sized smooth fruit that taste just like tropical kiwis, but without the fuzzy coat. Both Kiwi and grapes will produce best with proper training.

There are also various annual vines such as scarlet runner beans, sweet peas, and black-eyed Susan vines (Thunbergia) that you can plant between the perennial vines for a season or two while you wait for the perennial vines to get some size.

Unless you have a really narrow area, you might also want to consider shrubs planted close enough to the fence so that some of the branches grow through the fence to help hide it. Hydrangea paniculata such as Limelight, Quickfire, or Vanilla Strawberry might work for this if the area is at least 7' wide.