How can an adopted daughter atone for the "sin" of her adoption to her deceased father who raised her and treated her well, knowing her biological parents do not wish to communicate with her?

Do not affiliate yourself with the family that adopted you, and create a fictional name for yourself to avoid legal (Shar'i) complications. Make abundant supplications for those who adopted you. If the birth was a result of zina (fornication/adultery), then the man is not your father, but the woman is your mother, and you can maintain ties with her and show kindness to her.