Handling raw poultry turns my cuticles black!
As a student of medicine I've never heard of any condition similar to the one you describe, but then there are still a lot of diseases, conditions and genetic prepositions unknown to man (and to medicine students).
First I thought, it might be an allergic reaction to poultry proteins, but our skin normally doesn't have many immune cells on the outside (but can still result in a rash or excema) and you would probably have a really strong alergic reaction when eating poultry (that does not seem to be the case). 
What I would like to know is how long does your skin keep being black? Does your hands' colour change slowly or in a few seconds to a black? Do you have any other symptoms during contact to poultry or in general?

Anyhow there are few possibilities. Either something's really wrong with the meat you're buying (and I doubt that) or something's not alright with your skin. Maybe you have some kind of weird bacteria growing on your skin. There a lots of different bacteria physiologically growing on everyones skin ... stil this not seem normal to me. 
You should consider seeking professional help and talk to a general practitioner. 
Meanwhile I suggest using latex or nitril glove (you should be able to buy them in a drug store). This should prevent your hands from turning black and they still allow you to handle the meat precisely.