If I had to take a guess, it sounds like there's a problem with the seal on the little flap that covers the hole in the bottom of the tank.
Take a look at the inside of your toilet tank, from the flushing handle, there should be a chain that goes down to a rubber flap (about as big around as a tennis ball or so) on the bottom of the tank. Make sure that the chain isn't so long that the slack can get sucked into the hole. If it does, then the flap won't be able to keep the water in the tank, causing it to perpetually try and fill as it drains.
If the chain isn't overly long, then it's possible that the flap is just off-kilter or so old that it can't form a completely watertight seal any more, and will need to be replaced.
On the plus side, replacing the flap is pretty simple, and new ones are pretty cheap.