I think you already have your answer (look at the fill valve). When the fill valve stops working well, it continues to fill water into the tank even when the tank is full and the ball mechanism is pushed up by the water in the tank You can tell by looking to see whether the water level in the tank is right up to the edge of the overflow.
It is a very easy repair because you can install a new, inexpensive fill value, like this:
http://www.homedepot.com/p/Fluidmaster-Universal-Fill-Valve-400ARP25/100554467
The other thing that happens to me, sometimes, is that crud (for lack of a better word) gets on plastic part under the flapper (the flush valve) and prevent the flapper from sealing. Try wiping under the flapper with a paper towel.