Blackberry vines are notoriously difficult to eradicate. The key is persistence, and continuing removal treatments over several seasons. Unless the blackberries are caught and removed when they are very young, they will not go away with the first attempt at eradication. Completing only one step, such as cutting away the vines, can result in even stronger vines. Continue treatment until all of the blackberries are gone. Watch for small suckers that can reappear up to a year after cutting or removal.
Cut away vines and growth 6 to 12 inches from the ground using loppers or pruning shears.
Remove all the detached vines and branches, stacking them far enough away from the base of the plant that you have plenty of room to work.
Dig out each crown, the small mass at the base of each cane, that you can remove from the ground. It's not necessary to remove all of the roots as long as you can remove the crowns. This is labor-intensive, but it is the most long-lasting and effective method of removal.
Mow any remaining canes or branches down as low to the ground as possible using a weed trimmer or heavy-duty mower.
Rototill the area where the blackberry vines were located. Make several passes and remove as much of the root and crown material as possible.
Cover the area with heavy black plastic or garden fabric and a thick layer of soil and mulch.
Plant the newly cleared area with vigorous desirable plants immediately. The blackberries will very likely try to grow back, but the new plants will slow them down and prevent them from taking over again.
Repeat as necessary when new blackberry canes appear, or at least once a year. It will probably take two to three years to remove all of the blackberries permanently.