How do I improve the drainage of this grass driveway?
If it's heavily driven on, it's likely extremely compacted, which only increases the water run-off and retention (since it can't properly drain anywhere). 

can you use gravel? A gravel driveway would likely be the easiest solution.

Barring that, you could look at using a grid 'pavement' system:



It lets you plant grass without the turf having to take the brunt of the vehicle weight, so it remains porous.