A simplicial object in $Cat$ is a particular case of simplicial object, of course, with the $C$ there being $Cat$. It therefore has a simplicial set of objects and a simplicial set of morphisms. This is equivalent to giving an internal category in simplicial sets.
Remark A simplicial object in $\Cat$ is sometimes called a simplicial category, but that has other meanings.
simplicial object in Cat
One of the few references that discusses a model category structure on $Cat^\Delta$ as opposed to just $sSet Cat$ is
A model category sturcture on $Cat(sSet)$ presenting (infinity,1)-categories is in