How to revitalize old overgrown grape vine

I'm restoring an old garden; it is a lot of fun. There is a big old grapevine which has turned into an impervious mat of dead and live vines and various weeds and saplings. It is growing on and through an open-weave fence, and the whole thing is probably three feet wide by now. The fence has probably collapsed inside. How would I proceed? Fruit production does not have to be optimal.

Probably cut off dead vines. Check on fence's structure, maybe make a new one. Then train new growth up it. I wouldn't keep the large vines, you probably can't train them at this point. Anything woody should be cut off or trimmed for new shoots.  

I think you should maybe cut it down to one central leader.
I have done this with my vines, and they just come roaring back.
Now I have it trained along a cable spanning the north wall of the house, and I'm letting branches grow down from there.
Even though it will swine and grab, I tie it in place, to give me more control.

If the fence is metal, you might be able to burn off the growth, and recover the mesh.

most grapes I've had experience with fruit on 2nd year vine growth

Rule of thumb for cutting back any large/mature tree/bush/shrub is only cut 1/4 of it at a time, each year, into live wood.  

Cutting back the dead vines (although if it's a cold spring they may look dead, but just haven't leafed out yet) is obviously okay.  

Grapevines are tough, but they can still be shocked if too much is cut or there is a freeze/frost  on newly-cut wood, or hot, burning sun on newly cut wood where the cuts are larger than just pruning.