The old King has been murdered and a successor must be found, but first the killer must be found. The PCs, being members of the old Kings household, are tasked with the job of apprehending his killer.
Regarding the succession. The Throne is shared between three separate Noble Houses and the Crown cannot pass to the two members of the same House successively, thus it must go to one of the other two Houses.
Since nobody, not even an heir apparent, may benefit from the murder of another person, the succession cannot proceed until both contenders for the Throne, and all of their followers, are eliminated from the enquiry.
1. One of the heirs apparent, impatient for his shot at the Throne, assassinated the King in order to speed the process up. Being no fool, he did of course attempt to frame his opponent for the crime.
2. The Kings assignation was the work of a foreign power or major criminal origination that hopes to profit from the resulting chaos.
3. The murder was not at all political. Instead the old King, who always had something of a reputation as a womaniser, was killed by the jealous husband of his latest conquest.