You can purchase an online adoption kit, but it would be much better to have an attorney do this for you. It should not be very costly if the biological mother is willing to voluntarily have her parental rights terminated.
The problem with doing this yourself is that terminating a parent's rights to a child is a very serious matter, and the court is very careful and meticulous about making sure the parents rights are respected before he/she will grant the request.
An attorney can ensure that all of the technical parts necessary to ensure the parent giving up parental rights is done correctly and finally.
The last thing you want is to have a problem with the adoption to crop up AFTER the adoption has gone through, and now the mother wants her parental rights back and you begin that fight.
It is much better to get an attorney to handle this to ensure once it is over, the mother cannot come back and try to regain her parental rights - that would be a nightmare!
You certainly can use online forms, and in many cases they're perfectly fine - however, in this instance, my suggestion is to get an attorney to handle it.