In Texas, when a child is adopted, he becomes the child of his adoptive parents. He inherits from and through them. That means that the adopted child will inherit from the ancestors of the parent who adopted him as well as from the parent. There is no difference based on the adoption.
The question is often asked about the Texas inheritance rights of the adopted child from his natural (birth or biological) parents. Generally, the adopted child inherits from his natural parents and his adoptive parents. Of course, this only applies if the parents, adoptive or biological, don't have wills. If either or both have wills, their estate goes to whomever they say in their will. Texas does not have "forced heirship." An adoptive or biological parent does not have to leave anything to his children. But if he does not have a will and dies intestate, then his estate would be divided between all of his children whether natural (birth or biological) or adopted.
An exception to the above rule may occur if the parental rights of the biological parent were terminated by a court order. If the termination order is silent as to inheritance, then the above rule applies. If, however, the termination order says that the child will not inherit from his biological parent, then there would be no inheritance unless the biological parent left a will specifically naming the child as a beneficiary. FC 161.206.
An adopted child can contest a will of his parents just as any child can.