Performed upon the death of the previous leader, the Rite was overseen by the Arbiter of Succession. In 2367, Chancellor K'mpec took the unorthodox step of naming Captain Jean-Luc Picard as the Arbiter after his death, since no Klingon could be trusted.
The ritual was preceded by the elaborate Sonchi ceremony, which verified the death of the previous leader.
The initial phase of the Rite of Succession by the mid-24th century consisted merely of a proclamation that the two strongest challengers had been chosen, who would then fight eachother. In ancient times however, the Rite was a long, involved affair in which each prospective candidate performed the ja'chuq to prove their worthiness to lead the Empire. Picard chose to revive this procedure as a delay tactic while investigating an assassination attempt aboard K'mpec's attack cruiser. (TNG: "Reunion")
Even if only one challenger remained, like in the case of Gowron, this ritual took several months. The Rite of Succession's ultimate phase was the proclamation of the final remaining challenger in front of the High Council through the arbiter. (TNG: "Redemption")