Sphynx cats do not have hair to keep their skin free of oil. It can build up pretty quick and dirt then sticks to it. Here are some tips on how to groom them.
Run warm water in a sink or a tub, adding some shampoo to the water to soak their feet and nails.
Make sure you use a mild shampoo (tearless shampoo is always good).
Put the cat in the sink or tub and wet them down. Using a cup is always easiest. You will want to lather up your sphynx several times to remove the oil and residue that has built up. Thoroughly wash their feet and their butt. The feet get especially dirty from using a litter box.
Brush your cats teeth. It is just as important to brush their teeth as well.
Rinse well and swaddle them in a warm towel. (Keeping them well swaddled can help in the rest of the process)
Wipe their face gently around the eyes and ears with a towel or if they are more sensitive try cotton balls or cotton rounds.
Using a Q-tip clean out the inside corners of their eyes. They do not have eyelashes to keep dirt out of their eyes, so these must be cleaned. Be very gentle. This breed has very sensitive skin.
Using more Q-tips clean out the build-up in the ear, making sure you do not go too far in the ear canal.
While they are still swaddled, take one paw out at a time and clip each nail. Clean the nails and folds above the nails. There is a lot of build-up and dirt that accumulates there. Cotton rounds or baby wipes work well for this.