In California there is a streamline process for stepparent adoptions. If you have the biological parent's consent then the process can be fairly simple. If you do not have their consent it becomes a lot more difficult. You will need the court to rule in your favor. The easiest way to do that is to show that the other biological parent has abandoned the child. This is defined as not communicating OR supporting the child within the past year. If that is the case, the judge will often rule in your favor. If not the case, the stepparent will likely not be able to adopt the child.
Additionally, keep in mind that if the stepparent does in fact adopt the child that in the eyes of the law they are a parent with all rights to support, visitation, etc after a divorce or death. The biological parent's rights are essentially terminated and the stepparent steps into their place.