We bought our place before the Kudzu started greening up and I saw lots of vines and thought it might be the case but the seller made it out to be no problem. The way Kudzu propagates it is nearly impossible to kill it completely, the best you can do is control it.
Goats are one of the best methods if you can fence in the area. I had a local goat farmer that was interested in using part of my property but in the end decided not to because of a seasonal stream running through it that he thought young goats might drown in and some trees that the older goats may girdle and kill.
What I gave done to control mine: Mowing regularly if possible, I had a large area terraformed so it could be mowed once a month. I smothered one hillside with some old tarps I had that I spread out and weighted down with rocks and let the sun cook it for a coupe of summers. I also buried a bunch under shale fill where I wanted to extend a flat area over a drop off. In the deeper woods I noticed what looked like hoses running through the woods then up trees, turns out the Kudzu would make runs till it saw a little sun then go up. I spent a lot of time with clippers and bow saw cutting the vines to kill the Kudzu that was spreading out over the tops of the trees. Some were 2-3" in diameter.
Now for the real control method when you have areas you can not do any of the other methods. Trycera herbicide, I asked the people that control Kudzu for the power company to keep it from climbing the poles and ripping down the wires. They use a very strong mix with a truck mounted tank and sprayer and the Kudzu starts wilting with in hours. The active ingredient is Tricolor: 3,5,6-trichloroethane-2pyridinyloxyacetic acid 29.4%. I us a generic version called Crossroad from the farmers Coop. I mix it 8oz to the gallon which is about twice what the jug says but way less than the professionals. On a sunny day the Kudzu will be wilting by the next day. I used a lot of this the first couple of years with many heavy sprayings each Summer. Even hired a pro to do massive spraying to the worse areas twice. Now 6 years later I spray about 6 gallons (mixed) along a ravine that is not mow able and I do not own both sides of along my driveway and the edges of my woods. I do this level of spraying early when it is just getting leafed out and again in July. I do extra touch up spraying of about 2 gallons each time a couple of more times each Summer. Of course I continue watching and clipping new shoots or spot spraying as I see them.
I'm convinced that if I just stopped I would be back where I started within 5 or 6 years but the Trysera and mowing has it controlled. I started with it being solid Kudzu on both sides of about 800' of my 1200' drive, about 1/2 acres beside the house a couple of acres out in front, and hit and miss around the rest of our 7.5 acres.
Seeing what "bobsguns" posted while I was typing this long message, Roundup does not work well enough to bother with. The Trysera if used carefully will not kill grass and such but will turn it brown for awhile, it is a broadleaf herbicide.