I have an almost 7-year-old who loves to run too. He's very good at it and destroys all his friends anytime the distance goes more than 50 yards. His older sister (who doesn't seem to be gifted with running) is going to be running track this summer, but he's still too young to join, and he's very upset about that. I've run with him several times. The furthest we've gone is 2 miles, and he actually (because he doesn't know how to pace at all) throws in surges every now and then that I actually have to work a little bit to cover. It's crazy. I don't worry about wanting to "train" him at all, because when I take him to a playground, while most of the kids are playing on the equipment, he wants to go run to the fartest tree and back several times. He still likes to play on the playground stuff, but he'd much rather go running. I've taken him to the local high school track a few times, and he eats it up.
Just let your kid have fun and let him tell you when and how far he wants to run (within reason -- my son wanted to run a marathon this summer -- I said no).