We live Ohio, the Biological father gave up rights to his daughter in the 60's. She was adopted by the Mothers new husband shortly thereafter. Now the Biological father has passed away with no will, but left specific instructions with a witness for his wishes of the estate to be handled. There is no mention of his child in his estate, does she have any claim to his estate?

Almost certainly not. If she was adopted, she had a claim to her adopted father's estate, but under the law, she is not entitled to two bites of the inheritance cherry. If he had made a specific bequest to her, that is one thing, but she can't claim as his daughter since she is now legally the daughter of someone else.