What cause grime and dirt marks on the sink when one washes their skin with soap and water?

Dead skin/oils/dirt. The soap helps remove the dead shedding skin, the oils from the pores, and the daily environmental pollution that our skin is exposed to. This is the grime and dirt marks you are seeing in the sink.
Oil versus Water. Soaps in general are partially oily and partially water soluble. The soap surrounds a dirt particle, oily part pointing in and water soluble part pointing out as a smplistic answer. But it is hard to keep the combination completely disolved in the water and so some of it clings to the sides of the sink; if i remember my biochemistry classes at all!