Before we moved into our new house there were grape vines everywhere. We got rid of one stump, but another one by the fence is harder to get rid of. So I want to know what would be the most effective way to kill this plant off for good?
You seem to be implying that the stump is still producing growth rather than your just wanting to get the stump out of the ground. As 'stump' usually refers to a woody trunk of varying width, the way to kill it off for good is to buy a tree stump killer - in the UK, that would be something like SBK, described as a brushwood and tree stump killer. Method of use varies according to the product, but the usual idea is to drill into the stump, making holes about quarter to half an inch deep, particularly near the outside edges, all round if it is a large stump, then fill up the holes with the stumpkiller and cover with something - a large upturned pot or a sheet of plastic, anchored down, and leave it to die.
Years ago my neighbor and I spent most of the summer cutting, pulling and digging wild grape vines. When we were done we dug down around each grape root and we put about five lb of rock salt on each, covered it with dirt and watered it each day. In Oct we put another five lb on and to this day they have never come back. The best thing I ever found!