If you still have vegetation spray with a vine killer.

If you just have the root left.
Removing the root is the best way, but it may be hard with some vines which have a deep tap root.

I got at grape vine root out with a farm jack. See stump-removal.

How to kill it without digging it up.

• Cut it down to a stump and paint a herbicide on, which also prevent sprouts from coming up.
  Apply undiluted to freshly cut stumps, within 45 minutes. You should do it during the growing season.

  Some herbicides Are:

  • Bonide 2746 Stump & Vine Killer Concentrate - Use undiluted to kill stumps and prevent sprouting.
    
  • Ferti-lome Brush and Stump Killer Concentrate
    
  • "paint" the freshly cut stem with glyphosate (the active ingredient in most versions of Roundup) | UFL.edu. An Active Solution of at least 41% Glyphosate is required. Many Stores carry glyphosate at lower percentage active ingredient, but for stem injection you need greater than 41%.
• instead of painting a herbicide on the stump you can use a a herbicide injection gun or turkey injector.