You can have the driveway 'cut' for a 'gutter' at a strategic point closest to where it starts to 'run' towards the house. The gutter is something like 6-8" wide and about as deep as you need to make it, probably like 4", and a 'grate' is set on top so you can drive over it (and still catch the running water.) Maybe someone on the Forum will post back and tell you what they're called.
Anyways, of course, the angle will be such that the water is 'diverted' AWAY from the house. This means that if you have neighbors who will be the new 'receipient' of the water, you may not be able to simply just divert the water. IF the new flow happens to flood their property, they can sue (in most municipalities.) If this is a probability, then you'll need to continue with a water 'diversion plan' so it's channeled to either a very deep drywell or to another part of the property where it won't flood anyone, yourself included. If you end up burying some piping, be sure to put in clean-outs every 25' or so. A french drain might be in order but I can't say for sure because you say your soil is packed clay. If you post back, describe more about what's on both sides of your house/driveway and where else you might be able to divert the water. And lastly, tell us kinda where you live.