For kart track fun just about anything you build will be fun, unless of course you are an overly competitive type. The cost of wheels has dropped drastically since I first built mine and back then you saw a lot of creative setups. Wheels from streetbikes like CBR600 F2's, SV650's and EX500's were popular to budget minded riders. Mods to get them on were fairly straightforward and cheap. I probably would have gone this route if I had researched a little more before purchasing my wheels.
Brakes are very important as you will need far more braking than the stock setup allows. I warped several rotors trying to get by on the stock stuff. On my first decent setup I ran an EBC rotor and a custom machined bracket to relocate the caliper. I later swapped in a later model fork and caliper that was supported by the aftermarket. EBC has a fairly cost effective kit for most modern bikes that includes a caliper relocation bracket. The folks running modded street wheels usually fabbed up a bracket similar to mine or if a 320mm rotor was available for the wheel, Purchased relocation brackets from one of several sources.
Suspension wise a stock MX set up can be made to work reasonably well for kart track stuff, but most dual sport and trail bikes will need springs and valving work to feel right. They are simple set up too soft for agressive riding. The KLX 250 and 30 are nice little bikes and can go pretty good under the right rider. I would avoid a 2 stroke if I did it again. My YZ250 is a blast to ride, but no where near as flexible or managable as a 4 stroke machine. Don't get me wrong, it handles great and is very fast but I have to work a lot harder than the 4 stroke guys to carry the same pace and it is very poorly suited to street riding. Highly tuned 2 strokes like the YZ or WR simply do not like to crise at a set pace and can easily get dangerously lean doing so. Its not an issue on the track as you are constantly either accelerating or decelerating and never 'cruising' but on the street its at best annoying and possible very expensive. An XR400 is a good choice as well, but I have been considering an XR250r as a fun bike ever since riding one that a friend owns. You rarely need more power than what the 250-300cc trail/dual sport bikes offer.(unless you are that type A kinda guy...)