Pure water is the best and most universal cleanser. You can wash your face with pure oil, such as olive oil. Dampen your face, rub the oil in, and wash it off with a washcloth. It works perfectly, and is the best thing to use to get makeup off. Since I've been doing this my skin has been so much softer and clearer with more even tone.
You can use pure soap to wash your underarms or private parts but that's all you need it for. Borax is also excellent for washing the hands. It does not dry the skin and kills germs. You can also wash your hair with borax and rinse with citric acid as a non-toxic hair cleanser.
Dry skin brushing also cleans the skin well.
Most soaps are full of toxic ingredients. You can read more about non-toxic cleaners in Hulda Clark's book The Cure For All Diseases. Borax is also great for cleaning clothes. It is completely non-toxic and is odorless. You can even ingest it and it will not harm you, though it's not recommended.
Don't put on your skin what you wouldn't put into your body!