The Kilrathi Imperial family tree, as best I understand it.
The Emperor sires Gilkarg (who is executed for losing the Sivar) and the aforementioned uncle, who dies in the the first days of the fighting. During his life Gilkarg, as the son of the Emperor, is the heir apparent to the Kilrathi Empire.
Gilkarg sires Ratha and Thrakhath. At the time of Action Stations, Ratha would be the second in line to the throne (after Gilkarg), making him the heir presumptive, while Thrakhath would be third in line.
After Ratha is killed (or rather kills himself) in Action Stations, Thrakhath would become the heir presumptive. After Gilkarg's death at the end of SM1, Thrakhath would become the heir apparent (first in line for the throne), a position which he holds for the rest of his life. The Emperor ends up outliving him, but only by a short time
Gilkarg also sire a daughter, the Princesss and High Priestess Shukarra (sp?) whom we see in several WCA episodes. In addition Thrakhath sires at least one daughter that we know of from the novels. I have no idea where (if anywhere) the females would fit into the Kilrathi line of succession. There are also several cousins with claims to the imperial throne, as we see in WC2.
Best, Raptor