Full custody or Not?

I am having some problems regarding child custody and non payment if child support. I would like to file for sole custody of my 3 month old daughter, but I dont know what this would take. Her father is an active practicing alcoholic and very unstable. I hold a full time job, her father doesn't work and is never around. He has left horrible voicemails on my cell phone making threatning remarks and being horrible. I am the one finacially responsible for her, he has only given me $90 in support in 3 months. There is alot of history with his 12 year old daughter from another mother (whom he doesn't pay child support for either). I just don't trust him around my daughter. He is depressed and unstable. Do I have a case to file for full custoday and if so is it exspensive?