Euuw!! Keyplus's route isnt bad, but this is the alternative.
I drive pretty much that route every day.
Take the A312 from hatton cross, straight through feltham, towards the round about with the A316 (m3). Cross over straight over the roundabout, you will pass a big sainsburys on your right.
Shortly after that, you will hit a mini roundabout. take a left onto Park Road (A313).
Go straight on, down that road, straight on at the crossroads.
Continue straight on towards Teddington. You will pass an NHS walk in clinic on your right. Immediately after this, the road splits, you need to be in the right lane, and take the filter right onto Queens Road. Its a narrowish 20mph road. Follow that along till you get to a mini roundabout. Take a right, and then follow the road.
It will turn into a 30mph again.
Keep going, and eventually the road will curve to the left, and then back to the right, follow it round, then keep straight.
You will hit a T junction with no right turn(the way you want to go!) (A308) Hampton Court is sign posted from there -you will have to turn left and go around the Kingston Bridge Roundabout and back on yourself. Follow the A308, and you are there! Et voila.