1) About every 12th run, going by gut feeling. I had other drop patterns which were very similar, but not quite identical on the same map. At least one which started with a war staff, and at least two which started with a light plate. CK can be VERY consistent, and the shorter the total travel distance town - travi - CK, the more consistent it can get.
2) CK always drops one magical item (his primary drop, as an act 1 unique). Then he drops at least one, but up to five more, from his special drop. In normal, his primary drop can't upgrade to an exceptional item (which would then effect his special drop), so he is at his most consistent here. If you pay close attention, you'll notice that if he drops a blue wand, then his special drop might always be a war bow. Rare --> OK. Unique --> OK if you don't already have it. Set ---> Fail, and will drop next item, let's say crown. This will always succeed. However, if you have both the unique war bow and the unique crown, both will fail, and produce quintuple durability rares, and move on to the third item and so on.
Now, if he drops a blue ring, you would get another item as first item of his special drop; let's say a War Staff. Now, the same code runs, and if rare, OK. If set, also OK. If unique, OK as long as you don't have the unique war staff. If you do, he will drop the next item, which might be a ring. If you have all three unique rings, he will move on to item 3, which could also be a ring, etc.
You'll notice that the blue item always corresponds to his special drop. This allows you to create an excel spreadsheet where you record his blue item, then at least special item #1 with every run. There might be more than one special drop for every blue item, but not many. So if you have a short distance map, with the king not too well guarded by other cow packs, you can now use the spreadsheet to "investigate" which items will drop if you keep certain uniques. In normal, this is pretty easy, but in hell you could get 40+ different drop patterns even if you kill the king in apparently the same spot.
Anyway, there is a lot more to CK runs than meets the eye, and a lot less to it than reasonably random, like LoD
EDIT: Oh, and you don't always get 5 rares from a totally failed unique drop in NM and Hell. Pikes erroneously have their exceptional upgrade set to "Great Pilum" instead of Lance (which can't drop until 1.04). However, throwing weapons max out at superior quality, so if CK tries to upgrade, a Superior Great Pilum will drop instead. This will fail even the rare check (obviously), so he'll just move on to the next item if there is one.