I understand the application in a "disposable bow" like Fdeck's situation. If the hair really does have an excessive amount of "black finger gunk" on it, it isn't going to play very well. This is particularly a problem on French bows as your hand isn't in contact with the hair on a German bow. As far as excessive rosin goes, playing it out is the safest plan of action. A clean dry rag, or a clean dry toothbrush can also work wonders.
Using alcohol or anything of that nature worries me. If it gets on the wood it will remove the finish/polish on the bow. It is also dangerous around anything that is glued (the ivory/imitation/plastic tip, the eyes in the frog, the pearl slide etc.) as some of the glues used in bow making are not very strong. 99.9% Isopropyl alcohol is one of my go-to methods for removing glue in bow work, so it is a dangerous idea. If you go this route, or use any other products, make sure to keep them away from the stick and frog. From a hair point of view, alcohol dries it out making it brittle and susceptible to breaking. Although it would get some of the gunk out, it would likely do more harm than good in the long run.
My two cents is your bow hair doesn't really need special cleaning. If you use too much rosin, a clean dry rag/toothbrush is my recommended action. Chances are by the time an excessive amount of finger gunk has developed a rehair is in order. Try to avoid handling the hair as much as possible, but if you touch it once or twice it isn't going to explode or be completely useless. Hair takes a beating, stretches, breaks, and wears out. Any sort of cleaning method might help some, but ultimately isn't going to prolong its playing life that much.