Ah yes, kings. Gotta have kings in our stories, because they're the only ones in fantasy half the time that have the energy, the funds and the drive to start wars.
I have lots of different kingdoms in my world. Some are small and some are large. Some are too big for their britches and the rest of the world is afraid of the power they hold.
It is a good thing for the rest of the world that the two largest kingdoms have spent ages warring with each other. While this has kept them occupied, it has also become tiresome. The final battle is about to pick up the pace, but when one of the kings is defeated, what will happen to the rest?
If the tyrannical king wins, he's sure to just walk in and take over. If the other king wins, there will be peace. Or will there?
There are lots of reasons for kings to start wars. All you have to do is look back on history and mythology. Look at Helen of Troy. Fell in love with someone else and started a war because she left her husband. "The face that launched a thousand ships..."
Can you imagine starting a war over love? Men (as in mankind... not trying to be sexist!) have fought over lesser things.
If you want your kings to fight, pick a topic, any topic. It could be over a gambling debt, if you so choose.
Kings may be rulers, but it doesn't mean they are leaders. I have a few kings who are weak, who whine about being put in their position yet do nothing about it. I have kings who do do something about it. They range from compassionate to cry babies.
So depending on the king, you could have some fun with your world or you can make it miserable for all involved.