King's Court is a very swingy card. In a typical game, you will play KC - KC - copper-copper-estate, only to have your opponent play KC - mountebank next turn, or vice-versa. Scheme, however, eliminates most of the swing by allowing you to topdeck two king's courts and a scheme every turn. In the worst case, if you have no other targets for a king's court after playing KC- - KC - scheme, you'll still have a five card hand and three actions, and you'll still be able to return all three combo cards to the top of your deck.
If King's Court and Scheme are out on the same board as any cursing attack, this combo is a no-brainer. In the absence of discarding attacks, it may well be worth it to go KC - KC - scheme - scheme - (horrible attack) - (other horrible attack), playing scheme x3, schemex3, (horrible attack)x3, then (other horrible attack) x1.
This combo, as far as I can tell, has three effective counters. The most effective is minion, which seems to be the best scheme counter at any rate. Next up is possession - however, KC + KC + scheme is probably the best possible enabler for possession, so a possession board may well be a race to set up KC, KC, scheme, scheme, possession, possession. Last but not least comes the sheer opportunity cost of buying two king's courts, which is probably not worth it in the absence of +buy and good card-drawing actions. Militia is not a very effective counter: while it will cause a discard down to KC - KC - scheme, if your deck is at least 33% actions, which is not difficult to attain by repeatedly buying schemes, your second and third plays of KC will likely find targets, making KC-KC-scheme worth keeping.