Shut the water off to the toilet, leave it for awhile (overnight is good). Look in the tank. My guess is that the flapper valve is leaking. Depending on the toilet, it's usually quite easy to replace. Depending on your water and if you add any in-tank chemicals (don't do that!), they can last anywhere from a few months, to many years, but they do wear out - it depends on your water. Another thing you can do is put some food coloring in the tank, then see if you get any coloration in the bowl. It is almost certainly the flapper, and is an easy fix. The fill valve is just doing its job of keeping the water level up in the tank...it leaks, it fills. And, yes, if it gets bad, your water bill could be severely impacted.