It's going to be a tough job that will probably take a few years to totally eradicate it. You need to start cutting back all of the vines and find the main trunks of the plants. If they are growing up through trees, don't try to pull them out of the trees just let them die naturally once cut. Pull the vines out once they are completely dead and dried out.
Once you've cleared out the area and have the main trunks left, spray the trunks with a total vegetation killer then put a bucket or something that will not let light through over the stump. Watch closely around the trunk to see if any new shoots are appearing from the roots and hit them with weed killer. Make sure you get the tops and bottoms of the leaves.
Some people suggest digging out the trunks right away but I think doing this only encourages the roots to produce new plants. I used this method to eradicate honeysuckle and it took 2 years to finally kill the stump.
Not knowing what your zone is or where you live, it's hard to say how long it's going to take, but once you've found the main trunks of the plants and have removed the vines, the hard part is over but you must remain vigilant in looking for new growth.
