Here are the basics:
The KLR is a completely different bike. Different, more basic engine. A few shared parts, but not many. Turn signals, oil filter, just odds and ends are the same.
KLX-R is the trail version. No electric start, no secondary winding on the stator, so electrical system is basic. Has Nikasil lined cylinder. Both KLX engines are double overhead cams with shim over bucket valves. No doohickey, but the cam chain adjuster is still a sore spot due to poor design, Easily corrected. Plastic tank and seat are more dirt oriented than the C. Suspension is adjustable front and rear. Both versions have 43mm upside down forks.
KLX-C is the road version. Has a 1" smaller rear wheel, different swing arm, chain, sprockets, shock, seat, tank, instruments, plastic, lights, electric system, starter, cylinder has an iron liner so piston and rings are different. Head is the same. I think cams are the same. Crank and trannie are the same. Frame is very close to the same, but has different mounting tabs here and there. Performance is similar, but for the additional 30 pounds or so that the C carries and the slightly less compliant suspension and an inch lower ground clearance. Seat height is similar on the two models.
Any other questions....?