To start the adoption process, contact an adoption lawyer who's been recommended, as each state has different laws. If a public agency, such as the department of social services, is responsible for the child, contact that agency to say that you want to adopt the child. If your arrangement is that of voluntary kinship care, where you're related to the child under your guardianship, contact the child welfare agency. The child welfare agency usually prefers that kin caregivers adopt the children that live with them so that the children have a permanent home. Family members that adopt the children they've been guardians for may be eligible for financial adoption assistance.
Typically the next step involves a home inspection by the child welfare agency home to ensure the children will be safe and their needs will be met, even if the relative have already been caring for the children for years. The biological parents will have to voluntarily surrender their parental rights, or the court will terminate their parental rights if it feels that it's in the children's best interest. Finally, the adoption must be finalized in court, and depending on the age of the children and in which state they live, the court sometimes asks the children whether they agree to be adopted by their legal guardian(s).
A step-parent who's a legal guardian can also choose to adopt a step-child to protect the child from being sent to the non-custodial biological parent if the custodial biological parent dies, or to put all of the children in the house on equal emotional and legal footing (including inheritance issues). The non-custodial parent has to give up parental rights in order for the adoption to go through in these types of cases. If there's a positive ongoing relationship between the child and non-custodial parent, it might be better for the step-parent to just stay in the role of legal guardian instead of pursuing legal adoption (and an uphill court battle).