How can he get custody in Tennessee?

My fiance has a daughter from a previous relationship she is now 2 years old. The mother of the child has another child from a another man but does not have custody of this child, she lost custody of this one when it was born. My fiance had emergency custody of his daughter for the first 3 months of her life and he was attempting to get full custody but the judge ruling over the case was raised by a single mother and therefore partial to the mothers having custody. He has his daughter almost every day of every month but still has to pay child support. On the rare occasions that she is with her mother, she doesn't have a room to sleep in...is constantly moved from place to place because the mother doesn't have a stable home since the mother has a new fiance every month...is put in the company of men that do drugs....lives in filthy houses....and the mother never has money for food because she spends all the child support on herself to get her hair and nails done and get alcohol. My fiance and I would like for him to have custody because he loves his daughter very much and makes sure that she has food and is in a safe environment. He wants what is best for her but every effort he has made to get custody has been more and more disappointing and my fiance is getting a little discouraged even though he continues to try. the mother continues to threaten that she will not let him have his weekends and she threatens to take the child completely away and call police if he tries to see his daughter if he continues to try to take custody. I tell him she can't do that but I have never dealt with custody cases before and I don't know how to help him get custody other than give him my support and help him continue to not get discouraged....what can we do?