In the fall (approaching soon), cut the vine at the ground and paint the stump with a brush killer. Probably will have to repeat several years in a row.

Hello,

Just be sure it's grape vine and not bittersweet. You don't want to spread the seeds of bittersweet any more than the birds will by shaking and/or cutting it down in the fall unless it's early enough. We take care of bittersweet and wisteria vines for home owners and it can really be a handful! If it is grape vine you're all set, if not, repost with pictures!