$2-$4 limit O8 on stars, as usual these days. 5 handed.

I have [Kh][4d][2h] in EP and open raise. The cutoff reraises, BB flat calls and I call.

Flop: [5h][8d].

The cutoff is fairly lag; no clear read on BB.
BB checks, I lead out, CO calls, BB calls. Maybe a checkraise is better here?

Turn: . BB checks, I bet??, CO raises, call, call.

River: . Now BB leads out. Raise in order to secure half the pot? Or is it too likely I'm beat both ways?

Comments on other streets are also welcome.
Pieter

Pre-Flop
I raise there too.

Flop
I CR if there is someone that is agressive enough to bet. At a passive table I lead. I really want to get this HU and will do anything that will get me there.

Turn
Lead and call the raise. Depending on the quality of opponent I may 3 bet, but most of the time I call. Also HU I would 3 bet against most opponents.

River
As monk said you can't promote your hand because you are not going to make a stronger hand (the nuts) fold. It does look like you are beat in both directions so if the players are decent call and hope the 3rd player will call as well. If the BB is bad and the CO is decent I may raise to try and get HU against the BB but otherwise I call.

I'll be honest with ya, I don't like the hand much and usually have been folding those KKxx hands when xx is 2 low cards but not an ace. Problem is, you need an ace to hit to take low (usually) and when that happens your kings are no longer good for high. I'll play it shorthanded like you did, but I bet it's a loser over the long term in the mostly passive games that we're generally stuck in.

The lead on the river can be any flush in a shorthanded game like this, so I'd say your high has a better shot then your low, but I'd not be surprised to see you get boned from both of them. I still like your odds and will call hoping one of them has something like an AQ with a raggy low.