There are clans that are ruled by a High King, and this King leads them in the war against their enemy. The only way the king has power is because each clan has absolute trust in him. However the King all along was working for the enemy side and was trying to get the clans to destroy themselves. How would he get the clans to fight each other while maintaining his good reputation?
Take two clans, clan A and clan B. Bonus point if there is already a rivalry between them.
False flag attacks :
Dress some of your men as clan A men, and do some hostile stuff towards clan B like foraging or looting (a marching army need food, so it is a common practice to do it in enemy lands), or capture some nobles for some ransom.
Make them feel supported :
Tell the clan B leader that as you are in a war, you can not allow yourself to punish clan A, because you need all troops available, but that as clan B has been attacked loosely, you will tolerate a counter measure and not punish clan B hardly if they attack clan A.
From the point of view of clan A, who did not commit the crime, the counterattack from clan B will be unfair, and they will ask justice. Just tell the same speech as clan B.
You have now two clans, that both think that they have been attacked without reasons and think that they can reply with your support.
A bewildering array of activities can foster distrust...
Remember, most of these actions should be arranged by a third-party who is never seen in the castle. In fact, have your trusted lieutenant hire someone to hire someone. That way, you can deny everything if they screw up. Or when you have to have them killed to hide your tracks. These people are expendable. Might be a good way to get rid of folks in a third clan? Or empty your jails?
Also remember to have a solid network of eyes-and-ears to monitor the progress of your plots. they should operate in anonymity, unaware of each other or who they report up to. They don't report to you. They report to a local person who reports to a person who reports to your spy master. Again, plausible deniability.
1. foster competition
This is the most direct act, and should be first out of the toolbox. Maybe set up some sort of competition with rules so vague the competitors will inevitably fall into arguments over the actions of the opposing competitors. Directly set clan vs. clan in this competition. Make the winner's prize something trivial in the grand scale of events (a small bag of gold, bragging rights, the losers' armor and horse, etc.). Violent games work best, like jousts or boxing. Not officially "to the death," but if people die, all the better.
Hire some mercenaries to raid along a road between clans A and B. Send out double-agent spies to tell each clan that they are, in fact, brigands hired by the rival clan to hurt their trade.
3. Water rights
If there is a creek or stream that flows from one clan to another, send people to dam the creek near the clan border line. This robs one clan of the water their farms need. Of course, they will deny everything. Let this escalate.
4. Steal some livestock
Have some vagrant steal some sheep, horses, or cattle from a village in Clan A. Let them get caught. Be sure they have something incriminating from Clan B on their person.
5. Talk about it
Openly arrange private meetings with the head of one clan. Secret meetings. Talk about trivial crap; the meeting doesn't matter. The point is everyone will assume they matter a great deal. Wait a few weeks or months. Schedule similar meetings with a different, rival, clan.
6. Pass notes
Send notes from the capital, obviously forged, to clan leader A telling them to raid clan B. Send two copies of this note, one to leader of A and another on a path where clan B will just happen to find it. Let them figure out who's doing what based on these notes.
7. Tax hike
Hire some mercenary to go to clan B's villages and act as fake tax collectors. Plant evidence that they were hired by clan A. If, somehow, they aren't caught, then repeat the operation in reverse, raiding clan A. Eventually, someone will capture them and find the evidence.
Kill one trusted servant of clan A's leader, quietly, in the night. Be sure there is no evidence of who did this or why. Let the absence of data build up a life of its own; rumors will rush in to fill a vacuum.
Though distasteful, have a few servants of Clan B get kidnapped in the night. Leave them (alive or dead) tied up and badly hidden somewhere in Clan A lands.
10. Entertain me
Hire a band of travelling entertainers. Hide a couple of thieves in their midst. When the entertainers have moved on and the thefts are discovered, quietly let it be known that they were hired by the rival clan.
Send someone in to replace all of Clan A's wine with vinegar. (Or ale with urine. Or...) But don't alter any containers marked as Clan B's. Wait a few months. Do the same to Clan B.
If either clan sends out riders with messages, send professional pick-pockets to steal the mail. Read it. Plant altered messages that subtly shift their intent. Make sure the pick-pockets and forgers are good enough to not get caught. Watch as the clan leaders try to figure out why they don't get the responses to those messages that they expect.
Crusader Kings style of an answer.
Massacre one clan and disguise it as an accident (for example, make it as if a huge fire due to the dry season cause the whole village where the clan was living to burn, and it happened in an unfortunate dry and hot night) and make sure the clan head and his close relatives all died in that fire, which can be done by chopping them before roasting them. Afterall meat should be well cooked before consumed. evil grind
Now someone would have to rule over the land and the survivor. Who is this someone? One of the clans closest to the survivors? Or all clans closest to it will have the land split between them? Whichever you choose, this will break the power balance between the clans. It would be much much easier now to make the clans kill one another. Weaken one to lure other to attack. Make sure to target the content one and lure the more ambitious one. Afterall, power corrupts.
Now you just have to rinse and repeat with different methods of clan power reduction. Just make sure that in the process you never let one clan to have too much power, especially if it would match yours. Otherwise it might be you that becomes the target of the ambitious clan leader.
Appeal to tribalism support of clans. Explain to each clan why it's disenfranchized if one or more clans were to have power (especially if they are the larger clans) and that your rule is keeping that in check. Foster individual clan nationalism over kingdomwide nationalism, and refer to all individuals in all but a few large clans as clan-citizens rather than citizens. Keep the illusion that you will protect the smaller clans and ensure the same rights as all citizens by showing deference to small clan customs but not to larger clans, then treat the larger clans as being against the small clans when the demand you show deference to their customs.
Another fun way to achieve this is to show percieved favoritism to the larger clans while paying trusted agents in the smaller clans to point this out and stir up animosity in the smaller clans. Have your large clan base put on the defensive by countering any critism of small clan customs (legitimate or otherwise) as being anti-small clan bigotry and that the larger clans are hurting the kingdom by their unwillingness to work with the smaller clans. The King can further hasten this by making frequent proclimations, but keep them short and ambiguious enough to keep both the large clans and small clans seeing different messages in the statement. Also, have reasons for the small clans to see the king as a possible false king... perhaps he was a long lost heir apperent who only became heir apperent by appearing towards the end of his predecessor's life and was named as such mere months before the predecessor's passing left the throne vacated. This can be made even more effective if the heir presumptive was overall seen as little to no better than the new King (if only because said heir had a lifetime in the public eye compared to the out of nowhere heir), but at least was openly sympathetic to the smaller clans and acted like a monarch should (Again, out of years of experience when compaired to the usurper). It could also explain away loyalties to an enemy nation if the sudden appearence was orchestrated by the enemy nation in some way (such as kidnapping the current king early in his life and raising him to be loyal to them, before returning them to the country so they could reclaim the throne).