Reward him for successively closer increments of the wanted behavior. Meaning, the first few times, reward him for reaching a certain distance away from the string. Gradually decrease the distance so that he basically has to touch the string to get food. Because biting is an activity already connected to feeding, it'll be easier for him to learn to bite the string to get food (as opposed to doing something completely unrelated, such as rolling over). Then, after he starts biting the string, only reward him when he pulls on it, until he pulls it enough to ring the bell.
After he's got it down, only reward him sometimes, not every time he rings the bell. This is because rewarding him every time (which is how you trained him), a continuous reinforcement, has a very fast learn rate, but is also quickly abandoned once treats stop. Meaning all you have to do is ignore him a couple of times, and he'll stop ringing the bell at all. The best schedule is variable interval, because while it takes longer to learn, once they've got it, it's resistant to extinction (the behavior won't stop just because you don't treat). (Variable interval just means you reward at random times, despite how many times he rings the bell, so maybe the first time it's 6 seconds after he starts ringing the bell, the next time 30, the next time a whole minute, then 18 seconds, etc.)