It depends...
If you've defeated a kingdom as part of the main questline (see below*), you defeat a kingdom by getting the quest progress bar against that kingdom to '+100'. Infact, you don't even have to do it all yourself - other kingdoms beating them up will also move the quest progress. If you get to +100 against a quest-target kingdom, they will 'disappear' from Calradia; any remaining clans will move to other kingdoms (possibly including yours), taking any remaining settlements with them.
*If you've created/supported an imperial kingdom, the target kingdoms (I believe; never done it this way) will be Battania, Vlandia and Sturgia. If you've created/supported a non-imperial kingdom, the target kingdoms will be the 3 imperial kingdoms.
If you've defeated a kingdom not as part of the main questline - which seems to be your case - nothing. The ruling clan will carry on, much like a minor clan. It might even make a comeback. You can make peace with them, but because they've got nothing to lose, you might have to pay them for it.