Hi
You have to register an adoption deed in the registrar office of your area,if the biological parents are consenting. Some places the registrar will not agree without a court order.If the registrar makes objection you will have to get a decree from the court and the based on that court decree/order a deed of adoption has to be registered in the sub-registrar office. So first find out from the office if they need a court order as the biological father is agreeable and the custody of the daughter is with the biological mother.
If there is a divorce order , you should attach the same.
Biological father's presence is not needed only NOC from the biological father's is needed.
In the deed of adoption you can mention child' s new name with your name and surname.
Once the adoption deed is done give a gazette declaration along with a publication in local daily of the Changed name.
Since this is your step daughter, NOC from father alone is required. The biological mother can give you the child in adoption and she is the one party signing and you the other party, adoptive father.
After adoption the name in the birth certificate can not be changed as the biological father's name will be there. Name Change in the birth certificate is not possible, once it is entered have to be corrected through a court order. Since the facts are correct the biological father's name can not be changed in the birth certificate.
All other documents you can change the name as your name as father /guardian .
If you needed a court order , get a lawyer and proceed further.
Get NoC and adoption deed prepared by any lawyer .