What is the role of the British monarch? What role does the British monarch play as leader of the Church of England? Furthermore, how has this role changed over time?
The role of the monarch in the Church of England, while technically its head, is limited strictly to the selection of bishops.  Even here, involvement is limited, as the Monarch chooses from one of three candidates sent by the Prime Minister, from a list that was created by a synod of bishops.

Article 37 (of the 39 Articles) explicitly defines the role as follows:


  "The King's majesty hath the chief power in this Realm of England, and other of his Dominions, unto whom the chief Government of all Estates of this Realm, whether they be Ecclesiastical or Civil, in all causes doth appertain, and is not, nor ought to be, subject to any foreign jurisdiction ... We give not to our Princes the ministering either of God's Word, or of the Sacraments ... but that only prerogative, which we see to have been given always to all Godly Princes in holy Scriptures by God himself; that is, that they should rule all estates and degrees committed to their charge by God, whether they be Ecclesiastical or Temporal, and restrain with the civil sword the stubborn and evildoer ... The Bishop of Rome hath no jurisdiction in this Realm of England."


This makes him specifically not a priest, but as a sovereign with the prerogative of establishing ecclesiastical order, it does give him the power to appoint bishops.

One of the titles of the Queen or King of England is "Defender of the Faith," but Charles has publicly lobbied to change that to "Defender of Faith," so as to remove the last vestigial tie to Christianity. Initially, Henry VIII won this title for his apologetics.  The Pope revoked that Title in the 1530s, but Parliament conferred it on him via its authority.