Do i have a legit child custody of proving the mom unfit?

First you must ensure your name in on the child's birth certificate or you have no grounds for custody. Then file for sole custody and supervised visitation for the mother. Additionally, ask the court to appoint a Guardian ad Litem to the case (this is a court appointed attorney that works solely on behalf of the child by interviewing both parents, grandparents if they are available, siblings, other care takers, and they also make home visits to watch the parents interact with the child. If she poses any threat, is verbally or physically abusive, etc. they will report back to the judge in your favor). When you get your court date, provide the judge with the above mentioned items (make copies for your records) and ask the judge for a temporary custody order until he/she makes their decision. From the sounds of it, the mother's definitely unfit and you will be able to prove this quite easily as long as you have a legal claim to the child (biological father on b/c).