The Rite of Succession was a Klingon ritual that determined, between potential candidates, who would become the new leader of the Klingon Empire. The Rite was usually performed in the event that a Chancellor died without being challenged for his seat by another Klingon. The Rite was typically performed upon the death of the previous leaders who would appoint an Arbiter of Succession who would oversee the proceedings. The Rite was preceded by the Sonchi ceremony which confirmed the death of the previous Chancellor of the High Council of the Klingon Empire.
The initial stage of the Rite began merely with the proclamation that the challengers for the title have been chosen. This differed greatly compared to the old days where the process was a long affair with the potential rulers of the empire performing the ja'chuq which proved the persons worthiness as a leader. This part of the ritual became obselete and was rarely used though the death of Km'pec and the appointment of Captain Jean-Luc Picard, a Human marked the first time the ancient ways were used. (TNG episode: "Reunion")
Once this was completed, the candidates would compete with one another for the right to become the High Chancellor of the empire. Despite the removal of the Ja'chuq, it was still a lengthy process and take many months even when there was a single candidate. Once the challenger had defeated all his enemies, he would need to meet the Arbiter of Succession in front of the Klingon High Council. If there are no more challengers, the candidate is then chosen as the new Chancellor and leader of the Klingon Empire. (TNG episodes: "Redemption, Part I", "Redemption, Part II")
If a Chancellor is challenged to personal combat by another Klingon, and is subsequently killed the office of Chancellor immediately passes to the victor, or to the person designated by the victor. In that event the Rite of Succession is not performed. (DS9 episode: "Tacking Into the Wind")