He holds the highest ranking job of the feudal system, which means it is his responsibility to protect his entire kingdom - he answers to no one, but should he fail to protect his land, he could face anything from a revolution to complete obliteration by another king.
Which means, the two biggest things that preoccupy a King's time are:
- His People
- Other Kings
"His People" is an easy one - he has to order his lords to collect taxes and quell rebellion, while making sure everyone in his land is well-fed and productive. He can meet with a council, usually of Lords and Ladies, to discuss trade agreements, land disputes, large construction projects like roads and fortresses, and preventing them from jumping ship and joining with other Kings.
Given the demands that each individual Lord can have, and the fact that each Lord has an entourage of Knights that must be Knighted by the King, and the tendency for Lords/Barons/What-Have-You to be snobbish jerks, this alone can keep the King very busy.
This can all depend upon the specific setting you're using though, as some Kings hold direct authority over all their land, and don't answer to Lords at all. Or may employ mayors (somehow meshing feudalism with democracy). The important thing, though, is that the King is responsible for making sure the people in charge of everything else keep his Kingdom running without descending into anarchy.
Other Kings are a bit more tricky. A King can only rule so much land (unless your world is ruled by a single all-powerful King), and at some point he has to make some acknowledgement of other Kings surrounding him.
There are a number of ways that Other Kings can take up the King's time – They can negotiate trade agreements to better/worsen their economic position, they can declare war on the Kingdom and tax the king's military resources, they can throw their daughters at the King to get one of his Lords (or Ladies) to marry her and expand their own power, or they can just hold fancy royal balls where fragile footwear is all the rage. While the King doesn't have to deal with all of these issues directly, he does have to set policy for how to deal with the issues, and may occasionally be forced to meet with other Kings, or envoys from the King, or the other Kings' various annoying sons/daughters, or the other Kings' various annoying assassins or spies.
Which brings me to a third thing that occupies a King's time, but that really occupies everyone's time when they aren't an adventurer.
The King's Family
Unless the King is some kind of swinging King Bachelor (a possibility!), he likely has a wife, sons, daughters, and various cousins who all want part of his attention/power/crown. And like any father, the King has to deal with all of them. He has to find some way to occupy the time for his sons and daughters (does he send them off to war? Does he try to get them married? Does he let them while away their time in the castle and grow destitute? To say nothing of the Dragons that keep trying to kidnap them). If he is a bachelor, there's likely a slew of eligible young bachelorettes that are vying for his attention, which would also occupy a King's time.
That all being said... there is a fourth thing that is even less mentioned than the third thing.
Whatever He Damn Well Pleases
While all of the above are things he should be doing, as his responsibilities, the King is free beyond any other member of his class system to do anything else he wants in his spare time. He can go hunting in the forests, gallivant off with adventurers, declare war on his neighbors, marry another king's daughter, marry his own daughters (ew), study arcane magic, study combat tactics, raise taxes, lower taxes, paint a portrait, have a portrait painted of himself, play the piano, dance with his wife, dance with his husband, dance with his daughter (aww?), learn to cook his own meals, fish for trout, anything he wants to do he can do, and there's very little anyone can do to stop him.
...Unless there are people who could stop him, like a council who could get him dethroned, or an assassin who could spoil his good time, or a dragon who could capture him while he's on the hunt, or a rival king who doesn't take too kindly to him marrying his daughter, or a daughter who doesn't take too kindly to incest, et cetera.
In short, the King can do anything he wants... but has to face the consequences of anything he does, and has the weight of the world on his shoulders at all times.