I'm 10st 13 and 5'7", which makes me not exactly slim, but within "normal" BMI.
However, I cannot tell you how unfit I was. On the first week of the programme, I was panting after 40 seconds of running and it took me a good couple of minutes to get my breath back ready for the next 60 second run. Those single minutes seemed like FOREVER.
Seriously, if I can do it, ANYONE can.
It might be worth getting your back checked out, just to make sure you don't have any underlying issues there, but assuming you haven't, then I'd say:
Start off as slooooowly as you need to. Even if your "jog" is barely out of walking pace, if it's enough to get your heart rate up and slightly out of breath, then it's going to be enough to get you fitter. As you do get fitter, you find that your speed picks up naturally. My "run" was pitifully slow to begin with, but gradually I was finding that I was less out of breath and in fact WANTED to push myself a little more so that I was working a little harder.
When I started, a slow jog made me out of puff, and I needed the walk breaks to get my breath back. Now, I'm running faster and if I need a break I just run slower for a minute.
I've suffered with my back for years, but it's better nowadays than it's ever been - running strengthens everything. Just go at your own pace and listen to your body, really.