When did it begin historically?
The church covenant idea became commonplace with the Separatists (Brownists, etc.) in England (late 1500s). To the earliest Separatist a church was not a true, reformed church unless it had a church covenant that was adhered to voluntary (at least that was the position of the Scrooby, Leyden, and Plymouth congregations---and even the Salem church before John Cotton corrected them). In the words of John Robinson:
"…this we hold and affirm, that a company, consisting though but of two or three, separated from the world, whether unchristian, or antichristian, and gathered into the name of Christ by a covenant made to walk in all the ways of God known unto them, is a church, and so hath the whole power of Christ."
So is a Church Covenant a supplement to a subscription to the Confession of Faith of a congregation, or is it something different altogether?
Often a Separatist church had both a Covenant and Articles of Faith. The Articles were much like the Westminster, LBCF, or usually---among American congregationalists---it was based off of the Savoy Confession. The Covenant was more like an oath to one another to encourage one another, to watch over one another, to train up the children within their ranks, and to discipline if necessary. Both of those documents were usually "adhered" to by signing something, publicly owning it, or making a relation of their conversion that included a desire to "own the covenant".
Do the Presbyterians practice this and if so/ if not why?
Because Presbyterians hold to a polity that has synods, councils, etc. there is a more general "covenant", rather than individual church covenants for individual congregations (i.e. Congregationalism). Because Congregationalists---and Baptists---believe in the autonomy of the local church, they believed each church was only given legitimacy by having its own covenant.
For more I'd suggest Edmund Morgan,
Visible Saints; Perry Miller,
Orthodoxy in Massachusetts; or a more recent one David Weir,
Early New England: A Covenanted Society.