I'm what some refer to as a "shoe geek." In other words, I work in a running store and this is what I do... all... day... long... I know more about feet and shoes and biomechanics than is probably healthy for someone who does not practice medicine, so hopefully some of the following will come in handy:
-I will have to agree with Michael on having the proper shoes. The fitting process you described is pretty standard, but does not mean it was done correctly. If your feet are killing you within 5K, something is probably very wrong. You may want to go back to the store where you purchased the shoes and be evaluated once more. Most reputable running stores will work with you to make sure you are positively satisfied.
-Do your feet hurt first thing in the morning? Or when standing up after prolonged periods of standing? If so, it might be plantar fasciitis. The pain does not have to be strictly in the heel - it could be right up at the base of the toes! If it kills you to hop out of bed in the morning, that might be your cause.
-Over pronation can definitely cause some serious pain, so be sure you are getting enough (or maybe... not too much) medial support. Further, if the shoes feel unnatural, like the arch is too high or in the wrong place, that will cause some pain. You don't have to feel the arch to be properly supported, and something that feels "odd" doesn't mean it's good for you. Running shoes should be the very most comfortable shoes you own - fitting and feeling natural, like a pair of roomy but secure slippers.
-There is someting to be said for just getting started. When one is new to the running world, there are all kinds of new aches and pains one must overcome. Just as your calves or hamstrings can get sore, so can the muscles in your feet. If you are doing too much too soon, your body is going to tell you - and it might be doing so through your feet. Standard rule is increasing your distances by 10% (yes, only ten percent) per week. If you are jumping from 1K to 2K, that's a 100% increase, despite it seeming like such a small distance. If you look back at what you've done and you see some pretty dramatic increases in distance - and dramatic increases in pain at the same time - there might be a correlation. When in doubt, back off on the distances for a bit and see if your feet are a little happier.
That covers some of the most basic topics. Of course, there is a lot of other possibilities out there (running on softer surfaces, wearing synthetic - NOT COTTON - socks, etc), but limited means in which to convey them over the internet. If there is one last suggestion I might make, it is ICE after you run. Ten to fifteen minutes of ice can do your feet a world of good (be sure to keep something between your skin and the ice to avoid frostbite). And of course... stretch!