To strengthen the muscles responsible for this movement you will need to take a dumbell or a stick and place the weights at one end of it only. Holding the dumbell on the end that no longer has any weights on it, with your palm facing up, slowly rotate your palm so that it now faces the ground. Depending on your flexibility your palm may not be completely face down when you pronate your forearm. That is not a problem. Another thing to keep in mind is that only your forearm moves, there is no flexion or extension occuring at the elbow joint.
Supination is the opposite of pronation.
I have you do both directions because we don't want to promote muscular imbalances. Supination will help your backhand by the way. You will find it is easier to pronate than supinate with the same amount of weight. If you want you can split up pronation and supination so that you can use heavier weights for pronation. To pronate with a heavier weight you would stop the movement once your thumb is pointing skyward, then return back to the palm up position.