i agree with jan on the child support issue. you need to get it established and start paying it.
it sounds as if you are wanting to take this to court for modification. if you try to present the living conditions and you are not helping to support the child, it could backfire on you. i could see that she would have the excuse that "this is all i can afford. he doesn't help with support and i am trying everything." see what i mean?
you need to have proof of all the changes in residency. the child doesn't have mom and dad together, he needs something stable in his life. bouncing from home to home is definitely not stable. this would require some work on your part - snoopy doggy dogging is what my son likes to call it
you need proof of her residences, affadavit from land lords that she was a tenant, utilities, old room mates, etc.
you need proof that she is bringing men in and out of the home. judges are frowning more and more on live ins with children in the home. it isn't stable for the child if the CP changes mates often. any chance that old boyfriends would give affadavits? old room mates the same thing.
proof of the poor shape of the living conditions. unclean and unhealthy. constant drinking? drugs? anything of this nature. but don't start slinging accusations if you do not have anything to back them up with. it will make you look bad in the eyes of the judge.
you basically want to be able to show how your child would benefit if he were with you. you have lived in the same home for XX amount of time. it was the home were you resided with the mother previously. you have day care provided for when you have to work. that you have family that want to see him and spend time with him. that you are willing to work with her parents on spending time with him. (since they are basically done with her.)
don't totally trash the mom - after all, she is the mom. now, if you want to take action on this, you need to do it soon. it is hard to explain that you think things are so bad, but you have not acted on it.