From the bible we get an understanding of the priesthood of Melchizedek that, although God had established the line of Aaron to be the priest of Israel, He stepped in and added Melchizedek and his linage to a position above Aaron.
Comparing Jesus’ human linage to that of the recognized and acknowledged order of Aaron, He was not included. For the Jews, Jesus was not a priest, yet He claimed to be THE only priest that can take man into the presence of God.
The statement that Melchizedek had no beginning or ending refers to the absence of priestly descent and succession in his blood line, (by his birthright) and his appointment by God to a priesthood which has superiority to the Aaronic order.
There is no reason to believe that Melchizedek was a mystical figure. He, (like any faithful servant of God could be) was chosen by God to serve God’s people in a position he was not humanly qualified to do. His priesthood was a prophetic example of the priesthood of Jesus. The Lord was not raised in the Temple nor trained by a human father who had served all his life exclusively as a priest. His earthly father was a carpenter.
In the fashion of Melchizedek, Jesus was (and is) given a priesthood that is superior to the Aaronic priesthood. It had no beginning, and has no end.
I hope this sheds some light on this wonderful Old Testament Character.