First, identify the vine- so you can understand it's behaviour. Some vines like porcelain vines proliferate by self seeding. Some vines like wistersia send off tendrils and can actually strangle other plants. Some like boston ivy cling to bricks which can cause longtime damage.

Second, since you've decided to get rid of the vine, then look for it's main trunks, stems and start hard pruning. In fact, start pruning from anypart of the vine until you start seeing it's main stems. If it has taken over a tree, concentrate on bottom parts. Once you cut off the main trunks, the rest of the vine will die off.

Once you've pruned way, you will discover where it has originated from and from there start digging the roots out. Use some herbicides on the trunk only if necessary. I hesitate to recommend chemical approaches unless the situation is desperate.

Ianna