Extended Tevinter power structure chart ( expanded canon)
The Archon: Supreme executive position. Usually inherited -- can be elected by the Magisterium if they have no appointed heir (WoT I, p. 76). However, an Archon mustn't be a member of Magisterium or the Chantry (also based on WoT I). So, the choice between Maevaris and Dorian in DATV technically shouldn't have been permitted...
The Imperial Divine: Head of the Imperial Chantry, also holding the position of Imperial Grand Enchanter, elected among Grand Clerics. Has a seat in the Magisterium.
The Chanty and the College: not all enchanters in the Circles of Magi need to be Chantry clergy and vice versa, but all Chantry positions from Revered Mother/ Father upward require membership in a Circle (WoT I, p. 125). The Grand Clerics of the Imperial Chantry automatically take seats in the Magisterium.
(*Not canon compliant) The Imperial Chantry serves as the state religion, the Imperium's main moral authority, the main judicial and advisory body to the Magisterium. It provides checks and balances to the regular resurgences of the Old God cults and attempts to revert Hessarian's Transfiguration of the Imperium.
(*Canon Friendly): The Circles of Magi are the public academies, responsible for education, research and development of new thaumaturgic technologies that serve the public good.
The Magisterium: the main legislative body of the Imperium. Each of the Circles also appoints a representative, though the First Enchanters are not eligible (WoT I, p. 76). Other positions in the Magisterium are inherited. The Laetans class has an entirety of three Magisterium seats (WoT II, p.38). The Archon has the right to appoint magisters directly.
(*Canon friendly) A single person only holds one Magisterium seat. If a legacy Magister becomes a Grand Cleric, they hold their seat in the Magisterium as a Grand Cleric; the legacy seat is then freed to be filled in by the Archon's decree. The Archon holds a waitlist of Magisterium candidates who can "jump in" following emergencies and power shuffles.
The Publicanium (*not canon compliant): a secondary body to the Magisterium that serves as a lesser legislative body, and the main administrative body, that mostly operates on the provincial level. It includes both the central administration (e.g. the Treasury and the Imperial Highways department) and local governments.
If you want to picture how a single person can consolidate power by advancing in particular institutions, look no further than Magister Rezaren Ammosine from Absolution. Rezaren is a magister by inheritance and a Chantry Brother with realistic prospects of promotion within the Chantry, which also requires him to be a Circle Enchanter.
Conversely, Dorian Pavus has held an Enchanter position in a Circle when he aided Magister Gereon Alexius in his research, but he never joined the Chantry. From his example, we learn that a "fully fledged Enchanter" must graduate a Circle at a respectable level at least, and likely also pursue some academic work later.