Do I have any rights to inheritance?

It depends on (1) what the adoption decree states, (2) what the Will, if any, states, and (3), since your biological father is probably dead, how long it has been since his estated was probated. 

If you were legally adopted by another person, then you have no legal connection to your biological father any longer, and no, you would not have any rights to inherit as an heir. If your biological father (or anyone else in his family) decides to leave you something under a Will or trust, then you would have rights to that as a beneficiary. However, an adoption normally cuts off all legal relationship between the adopted child and his or her family. The adopting parent and the adopting parent's family become the adopted child's new legal relatives, and it is only from your adopted parents and their families that you would have the right to inherit.   If you were adopted by a stepfather, then normally you would remain legally related to your mother's family, but your stepfather would become your legal father after the adoption, and you would then be legally his heir, not an heir of your biological father.