I would imagine a KLR650 would be a decent first bike. Although I have never ridden one, so I can't speak with 100% certainty. But, I do have a Suzuki DR650 which I picked up a month ago. So I am comparing the KLR to my DR. The KLR is more street oriented, where as the DR is more dirt oriented. But both are similar bikes which get compared to each other all the time. So consider a DR when looking. I was open to a KLR, DR or Honda XR650L when I was looking. I worked out a trade deal on a DR, so it is in my garage.
It will be a taller bike than a dedicated street bike, but that shouldn't be a problem for you. In fact, it will give you more leg room for more comfortable long distance riding. It will have enough power to keep you interested for a year or more, unlike a small 250. I have ridden liter bikes (Suzuki Bandit 1200S, Yamaha FZ-1, Honda CBR1100XX) for the past 15 years and the DR has just enough go to keep me interested in it. If you like riding, you will be interested in getting a larger street bike eventually like everybody else does. But a 650 single is just enough to be a road bike that you can take on the highway. So it is a bike you can do a bunch of riding with on different roads (highway, secondary back road, dirt roads, etc.). If you are exclusively going to ride secondary roads, you may also consider a DRZ400. I live in Phoenix and all the good riding is 45 minutes to an hour away via highway, so I went with the DR650 over a DRZ400.
As mentioned, if you buy a cheap KLR, you can ride it for a year and not loose much when you sell it. So as long as you don't destroy it by crashing or lack of maintenance, it is a great way to get into biking to see if you like it. I definitely recommend getting a used bike as they are cheap and you will most likely crash it in the first year. So no need to dump a new bike when you can do it to a cheap used one. In fact, don't pass up a previously crashed bike, they can be had cheaper than a non-crashed bike. And you won't feel horrible when you crash it again.
I also can't stress enough how important good gear is. So take the cost of gear into consideration as an up front cost. Helmet, jacket, pants, gloves and boots will help protect you in a crash. And as a new rider, your chances of crashing are much higher than an experienced rider. Gear can make a low speed tumble into something you can walk away from instead of getting hurt. It can turn a medium grade crash into something you limp away from rather than taking an ambulance ride. And it make a lethal crash into something you can survive.
Also consider a new rider's course. I didn't take one and regret not doing so. I do think it could have taught me in a few days what took months or years to learn on my own. Plus it satisfies you motorcycle license requirement and gives you an insurance discount.