There is sufficient evidence to show that Melchizedek was not even an Israelite, but a Canaanite who worshiped the true and living God. He would have done this apart from meeting Abram, who actually came to meet Melchizedek from a polytheistic background!
There is no record showing that Melchizedek received his priesthood from anyone. Furthermore, there is no record indicating the particulars of his birth or his death. Being a man, he was certainly born, but his ministry did not require these vital statistics. Whatever his priesthood was, it was historical, genuine and continues without interruption.