You need to go to your local electronics store and get some contact cleaner.
To clean the pots, remove the screws from the bottom of the joystick base (should be 8 screws in total, four outer screws and 4 inner screws). Remove the base and be careful to keep the joystick inverted to keep things from falling out. Remove the wire tips from the potentiometer contacts (REMEMBER WHERE EACH WIRE GOES!!!) and with a Q-tip, dip some contact cleaner on it and gently rub it on the potentiometer contact. Do that with each of the potentiometers. Once you are done, make sure that you have the wires in the correct positions and put the base back on. That's all there is to cleaning them.
If you find that the potentiometers are bad and need to be replaced, your best bet to find new ones is not through the CH Products website. They charge $10 bucks for each potentiometer. You can do a Google search for CTS Potentiometers (they are the ones that make them for CH Products) and buy the 100k ohms CTS series 295 Rotary Potentiometer in bulk. This way you can keep your joystick alive for years.
Ack-Ack