You have to contact family court in your area. The process is lengthy. First, they will have to try to terminate the biological fathers parental rights- to do this, they have to contact him and let him know what your intentions are. If he refuses, he can be held liable for any and all back child support- that alone is usually enough to scare a non present parent from fighting it. If he does refuse, and has been paying child support all along, then you can take him to court to have the rights terminated without his consent- having not been around the children is going to be enough for the court- especially if the intention is to have the children adopted by your husband- who is basically their dad.
My son also has a non present biological father. And then he has a Dad (my husband), But I do not want to terminate the biological fathers rights, because if I do, he will not be required to pay child support any longer, and I feel it is his duty- I sure didnt get knocked up by myself.
If the father in this situation is paying child support, and you want him to continue, then you cannot terminate his parental rights- but what you CAN do is legally change the childrens last name- you do not need the permission of the biological father to do that. I assume the children already call your husband "dad"- so if you change the childrens last name to match your married name, who would know that they arent your husbands children? A good reason to change the childrens name is to make school registration easier, the court accepts that as enough cause.
Ok, then, if you successfully terminate bio's rights, then you need to apply for adoption.
In the end, it just seems easier to change their name I think.