The Mormon teaching on the Mechizedek priesthood has absolutely NO BASIS in the Bible. Joseph Smith claimed that he received a revelation about this, therefore he laid out the following teaching in Doctrines & Covenants, Section 107. So this is a purely Mormon invention.
SECTION 107
Revelation on the priesthood, given through Joseph Smith the Prophet, at Kirtland, Ohio, about April 1835. Although this section was recorded in 1835, the historical records affirm that most of verses 60 through 100 incorporate a revelation given through Joseph Smith on November 11, 1831. This section was associated with the organization of the Quorum of the Twelve in February and March 1835. The Prophet likely delivered it in the presence of those who were preparing to depart May 3, 1835, on their first quorum mission...
1 There are, in the church, two priesthoods, namely, the Melchizedek and Aaronic, including the Levitical Priesthood.
2 Why the first is called the Melchizedek Priesthood is because Melchizedek was such a great high priest. [Note: There is no reference here to Christ, and Melchizedek was not a High Priest, just a priest. Christ is the High Priest]
3 Before his day it was called the Holy Priesthood, after the Order of the Son of God. [Note: This is the reverse of what is in the Bible]
4 But out of respect or reverence to the name of the Supreme Being, to avoid the too frequent repetition of his name, they, the church, in ancient days, called that priesthood after Melchizedek, or the Melchizedek Priesthood. [Note: This has no Scriptural basis]
5 All other authorities or offices in the church are appendages to this priesthood. [Note: This is a purely Mormon belief]
6 But there are two divisions or grand heads—one is the Melchizedek Priesthood, and the other is the Aaronic or Levitical Priesthood.
7 The office of an elder comes under the priesthood of Melchizedek. [Note: This has no Scriptural basis]
8 The Melchizedek Priesthood holds the right of presidency, and has power and authority over all the offices in the church in all ages of the world, to administer in spiritual things. [Note: This is a purely Mormon belief and everything that follows]
9 The Presidency of the High Priesthood, after the order of Melchizedek, have a right to officiate in all the offices in the church.
10 High priests after the order of the Melchizedek Priesthood have a right to officiate in their own standing, under the direction of the presidency, in administering spiritual things, and also in the office of an elder, priest (of the Levitical order), teacher, deacon, and member...
15 The bishopric is the presidency of this priesthood, and holds the keys or authority of the same. [Note: This has no Scriptural basis]
16 No man has a legal right to this office, to hold the keys of this priesthood, except he be a literal descendant of Aaron. [Note: This contradicts verse 15)
17 But as a high priest of the Melchizedek Priesthood has authority to officiate in all the lesser offices, he may officiate in the office of bishop when no literal descendant of Aaron can be found, provided he is called and set apart and ordained unto this power by the hands of the Presidency of the Melchizedek Priesthood. [Note: This contradicts verse 16]
18 The power and authority of the higher, or Melchizedek Priesthood, is to hold the keys of all the spiritual blessings of the church—[Note: This has no Scriptural basis]
19 To have the privilege of receiving the mysteries of the kingdom of heaven, to have the heavens opened unto them, to commune with the general assembly and church of the Firstborn, and to enjoy the communion and presence of God the Father, and Jesus the mediator of the new covenant...