You haven't said how old you are. I think that once you are 18, you can do whatever you want as far as contacting your father.
Second of all, you might want to go to a counselor. Again, I don't know how old you are, so if you are in school, talk to your school counselor, and ask them where there are psychologists/mental health counselors.
In no way am I suggesting that you have mental problems, but if your mother is abusive, that abuse causes mental problems later in life. Once you talk to a therapist and you guys find some solutions, maybe your mom can go to the therapist with you, and you can confront her about hiding all the letters, and trying to control the situation.
Depending on how abusive your mom is, do you think maybe you could sit down with her right now and have a talk? Ask her calmly if she feels afraid that you might leave her if you form a new relationship with your birth father. Parents do a lot of controlling things, just because they don't want to lose you.
On the other hand, from what you say, you still don't know your father really well, and I wouldn't recommend trying to go live with him until you know him much much better. There COULD be actually a good reason why your mom doesn't want him contacting you. This is another question you should ask your mom (and hope she is honest with her answer): Why doesn't she want you to communicate with your father?
Good luck.