What Apostles-Creed affirming groups claim priesthood of Melchizidek? I was looking at the online biography (perhaps an AOL or MySpace biography) of the friend of a relative some 20 years ago.  I was alarmed that my relative was associated with this person, based on some very immoral vices the person was proudly describing on their page.  One point that shocked me was that this person said they were "a priest of the order of Melchizidek."  It was the only religious statement on the page.

Q: Do any orthodox Christian groups (perhaps defined here as affirming the Apostle's Creed) use such language to describe themselves? Or would this be some sort of cult?
The LDS Church uses this phrase.  They divide the priesthood into Melchizedek Priesthood and Aaronic Priesthood.

Melchizedek Priesthood is described on their site as follows:


  “There are, in the church, two priesthoods, namely, the Melchizedek
  and Aaronic” (D&amp;C 107:1). The Melchizedek Priesthood, which is “after
  the Order of the Son of God” (D&amp;C 107:3), is the greater of these. It
  “holds the right of presidency, and has power and authority over all
  the offices in the church” (D&amp;C 107:8). It also holds “the keys of all
  the spiritual blessings of the church” (D&amp;C 107:18). It is named after
  a great high priest who lived during the time of the prophet Abraham
  (see D&amp;C 107:2-4; see also Alma 13:14-19).
  
  The offices of the Melchizedek Priesthood are Apostle, Seventy,
  patriarch, high priest, and elder. The President of the High
  Priesthood is the President of the Church (see D&amp;C 107:64-66).
  
  Men in the Church must be worthy Melchizedek Priesthood holders in
  order to receive the temple endowment and be sealed to their families
  for eternity. They have the authority to administer to the sick and
  give special blessings to family members and others. With the
  authorization of presiding priesthood leaders, they can bestow the
  gift of the Holy Ghost and ordain other worthy men to offices in the
  Aaronic and Melchizedek Priesthoods.
  
  When a man receives the Melchizedek Priesthood, he enters into the
  oath and covenant of the priesthood. He covenants to be faithful,
  magnify his calling, “give diligent heed to the words of eternal
  life,” and “live by every word that proceedeth forth from the mouth of
  God.” Those who keep this covenant will be sanctified by the Spirit
  and receive “all that [the] Father hath.” (See D&amp;C 84:33-44.)


Specifically, John Taylor is quoted as saying  (Emphasis mine)


  Seventh—That while the power of the higher, or Melchizedek, is to hold
  the keys of all the spiritual blessings of the church; to have the
  privilege of receiving the mysteries of the kingdom of heaven, to have
  the heavens opened to 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, and to preside
  over all the spiritual officers of the church, yet the presidency of
  the high priesthood, after the order of Melchizedek, have a right to
  officiate in all the offices in the church, both spiritual and
  temporal.