My father is adopted and has been diagnosed with terminal cancer. His adopted mother is alive who is next of kin me or her?

The first person would be his spouse, then his children, then his grandchildren or great-grandchildren.  The adopted mother is an heir only if there are none of these. The estate is responsible for funeral costs, not any of the heirs.  But from a practical standpoint, the heir would pay for the funeral and then be reimbursed by the estate. If you dad passes, then his heirs are entitled to all of his property.