Kulkarvek was the only Urgal king. After his death, his body lay in state in the Barrows of Anghelm. Of the warnings that the Umaroth gave Murtagh and Thorn for their travels in the north, this was the first. The quote from Umaroth is as follows:
"We know much of the difficulties you have faced, Thorn and Murtagh, for we have watched you from afar, even as we have watched Eragon and Saphira. There are many things we would teach you once you are ready, but until then, we will tell you this: in your wanderings, avoid the barrows of Anghelm, where the one and only Urgal king, Kulkarvek, lies in state... Do this and, unless you are greatly unfortunate, you shall not encounter danger beyond your ability to master."
This could mean that, with all the immense abilities Murtagh and Thorn had gained from Galbatorix's meddling in their minds and bodies, as well as even their knowledge of the Name of Names, they would still not be capable of killing Kulkarvek if the Urgal King were to wake up from his "slumber". Kulkarvek can therefore be presumed to be incredibly powerful, perhaps on par with the late Galbatorix himself (when the latter had been enhanced by the Name and numerous Eldunarya), and since the Kull are the strongest group within Urgal society, it is presumed by most fans that Kulkarvek is a Kull. However, without further evidence, speculation on the matter proves fruitless.
When asked if Kulkarvek would play any role in future stories. Paolini only replied "Maaaybe. :D"