I am 21 years old and my father has never paid a cent to my mother for child support. He has a good paying job and every time I ask him for a little cash to help me out he says he doesn't make enough to pay me anything. Would I be able to file support against him?

Nope.

Child support is for the maintenance of a child--and is paid to the custodial parent.

Your MOTHER has to file for back support.

No, you have absolutely no right to collect child support from him. That is something that your mother would have had to file for while you were still a minor. If she did not file for it while you were still a minor then he owes her nothing. If she did get an order for support and he just never paid as ordered by the court then she needs to go back to court to have it enforced. There wouldn't be any more current support but whatever was in arrears could still be collected.