My preference for the back woods is a 250 cc with nobby tires and no fancy accessories. These bikes are light enough to upright, will go through anything if you use a bigger rear sprocket, and can be found in ample supply at affordable used prices. Most are very durable and can sit for long periods if you run the gas out of them after each trip out. They are also easy to trailer.
If you are really looking at dirt, then IMHO a 200 - 250 class bike is probably the way to go. The WR250R, XT250, CRF250L KLX250, the older 225 / 230 Yammie / Hondas and so on are all good dirt bikes for the average rider, and more than capable of road commuting or even light touring use, as long as you don't expect to do lots of interstate.
The KLR650 is a good bike, but as a real dirt or city commuter, it is at least 100lb too heavy, as well as being too wide.
THey also do not crash well, particularly the 2008 plus model, a drop in the dirt can work out pretty expensive.
A DR400 is a good choice as well, or for an off-the-wall choice, something like the Husky TE610E can be found used very reasonably, has a lot more character and zip than the KLR, has much, much better suspension, and weighs under 330lb fully gassed (KLRs cross 450 lb).
Wide ratio 6 speed box and much more power, and better quality suspension move it ahead of the DRZ, but you need to stay on top of maintenance. Service intervals are 3000 miles, but you really do need to check things like valves and camchain tensioners (BTDT).
Lots of fun older bikes around, like the DR350, XR25/400 (if you can get one plated), XR200 and so on.
For fun in the dirt (or commuting dings and knocks), crash damage resistance is one thing that is often ignored. I high-sided my TE610E on a dirt road at about 30mph, the bike landed on upside down on the front fender and LH handlebar / LH side. I picked it up and rode for the rest of the weekend, with a tweaked handlebar,
Ignoring the scrape in the front fender, and a few other minor marks, it cost me under $150 to put back into riding position. $85 of that was getting the aftermarket HID headlight unit repaired (MR16 'low' beam projector was damaged). A new aluminum handlebar cost $30 at the local ATV shop.
A DRZ400, DR350, DR650, XR650 and so on will probably crash just as well, if not better.