I'm a mother of 3 all under 7yrs old. I want to take their biological fathers right away. Who in mind does not see them or calls them. Is a fight to even get help from him. He is currently mentally unstable and to the best of my knowledge is on drugs. Mg question is where do I began to start the process and is it a possibility that I achieve my goal.

What state, if the US?
Is he on their birth certificates?
If not, has DNA testing been done on all 3?

You start at Family Court by asking them what their process is and what forms are needed. It might be good to go there in person and talk to the clerk.
The entire process may be slow, because the father has some rights, and generally must be contacted to see if he wants to be involved.
You can't just say you hear he's on drugs - you need criminal records.
Keep in mind that applying for full custody may make him decide to fight it, so be prepared for protective measures.

If you have a social worker, start there.

Forget it! A court will be very reluctant to terminate a father's rights. Even under the circumstances you cite. A court is going to say he is not exercising his rights now so why take them away? A better option is to get yourself awarded Full Legal and physical custody with no visitation (or no more that supervised).

There are many threads here (I've moved your post from the Children forum) including some sticky notes at the top that discuss this issue.

He lives in new Mexico and the Children and I lived in Oklahoma. He is on two birth certificates. No DNA test have been done. One child that his name is on has a great possibility of not being his. He knew that when he asked to sign her certificate. The other two is no doubt his. His name is on one of the twos certificate. Not on both because he refused to sign his when he was born. Due to me being 16 at the time and he was 20 although he was at the hospital and stayed. If I asked for a drug analysis test would he have to give one in court? And if it proved he is doing drugs will it make things easier for me?

No, he merely being a drug user will not get his rights taken away. It just does not happen. You may get full custody, (sounds like you already have it) since he does not do anything.

He will have the rights to go back to court later and ask for visitation.

If he was in jail for long term, some courts (not all) may take the rights away. Or if he was a threat to the children, (but even then courts may just require supervised visits) The taking away of rights is a myth.

If there is other evidence of drug use, you may request that the court order a test.

Were you ever married to this man? Are there any existing court orders for custody/visitation?

Frankly, I think you should save your money. It sounds like he has nothing to do with your children now. So why spend money? As I said its unlikely a court will terminate his rights in any case.