There are many ways to go about it. Depending on the size of the kudzu patch, repeated applications of Round Up (glyphosate) or Brush B Gon (triclopyr) is the best way. It will however take repeated applications in order to kill it. Spraying it now in the fall is a good time of year because it will move more herbicide into the plant. In the spring, apply after it has good leaf cover. This is usually about May. Allow it to put out new leaves after the first application in May and apply another application of herbicide. Eventually you will wear out the plants and they will give out of reserves and die.
Another way to go about it is to have it grazed by goats. Now grant this is only possible if the patch is fenced in. There are companies that will rent goats to landowners for this very purpose.