The best way to clean a dirty video game controller minus the water sensitive electrical components is a combo of hot water and dish soap, preferably Dawn dish soap due to its stronger ability to cut grime and dirt. Just place the plastic parts in a small plastic tote with hot soapy water and proceed to clean each part with a clean paint brush or a palm dish brush. Be mindful of any small buttons when you drain the tote.
Then place plastic components into a colander and rinse it off under the faucet with hot water to clean the soap off thoroughly. Either dry the components with a micro fiber cloth or paper towels. Piece the controller back together and enjoy a clean gaming controller.