This campaign is described in terms of a more or less medieval European fantasy game in system agnostic terms, but it can easily be adapted for use in most genres.
In his day the King was a great hero, crafting his empire with steel and strength, magic and sheer willpower over the course of his life. He settled down, like all great men do, married, and had five children. While he had himself grown up in poverty and strife, his children wanted for nothing, having the finest of everything.
On the day his youngest son turned sixteen, the king grew troubled. It took strength to unite a kingdom, strength to keep it together. It took a great man. History was littered with the ruins of empires that fell when their founded died, torn apart by the bickering of heirs.
That very day he called his children together to tell them he had chosen an heir.
He would not see the kingdom divided upon his death, and so he told his children that when the time came, only one of them would rule, and the others subordinate to that ruler's will. The heir would be chosen simply: whomever had accomplished the greatest deeds, become the greatest, in five years time. The king would judge their accomplishments, and they would be supreme.
His children protested, of course, until the king reminded them that the contest had begun.
The scattered to the winds.
Now the five offspring travel the kingdom, pursuing greatness. What exactly that is the King refuses to elaborate on, but most bookies assume that since he was an adventurer, it must mean that heroic deeds are the order of the day. Others insist that the king has mellowed in his old age, and greatness is power, so whomever builds the greatest power-base is the most worthy.
And, of course, each of the prince's children has their own opinion.
A lot of people stand to gain or lose depending on which Prince is victorious, and not just gamblers. Different factions - craftsmen, the church, the wizard's guild, the armsmen, and so forth - each has a stake. And any of them might interfere or participate, to help or hinder any of the princes in the contest.
I've left a lot for individual GMs to define, to make this campaign usable in almost any setting. Questions you must ask:
Who are the princes? Define them. Their strengths, their weaknesses, their personalities.
What is it the king really wants? The players might not know, but you should. Maybe if they investigate his past, they can find out. That would be priceless information.
What factions have a stake in this, and what are they willing to do to get their way?
How do the players fit into this? Are they agents for one of the princes? Maybe one of the players is an heir. Or maybe they're secretly helping one, or hindering another. Maybe they're agents for the king, spying on his children.
After you know what you're doing, your next step is to map out the progress of the contest. Decide how the NPCs will fare, what they will attempt, how they will interact. The game itself is a series of opportunities for the players to impact the contest for whichever agency they serve.
Maybe the players are foreign spies, either trying to help the weakest prince win, or to ensure that none of them do, so that the king dies without a clear heir.
Maybe the players are gamblers, trying to "fix" the contest for personal gain.
Maybe the players are the kings agents, trying to keep his children from cheating.