You have already checked the water supply valve and water line so we can rule that out and now concentrate on the tank fill valve.
Question,,,is any water entering the tank? If so, check the flapper valve in the bottom of the tank, it may be stuck open and the water is draining out as it fills. Solution- reset the flapper.
If there is no water entering the tank take the lid off the tank and check the float mechanism. If the float happens to be stuck in the up (Full tank) position the valve will not open. Press down lightly on the float and it should fall to the bottom and the valve open. If that is not the problem it is an internal problem in the float valve.
If you have the old ballcock style fill valve it is often very difficutl to get the valve body open to service, and equally difficult to find replacement parts. The simple solution is to remove the bad fill valve and replace with a universal replacement fill valve such as a fluidmaster fill valve.
The universal replacement valves can be bought at any hardware or home supply store for about $10 in a complete kit. Installation is very simple and straitforward, requiring only a pair of pliers and about 15 to 20 minutes time.
If you already have a Fluidmaster replacement fill valve you can get a replacement rubber disphram for about $1. To replace the diaphram, turn the water off at the wall stop, pry the black cap off the top of the fluidmaster fill valve, lift out the white valve mechanism and remove the old rubber diapram. Insert the new diaphram (Carefully note the needle valve- it goes through a small hole in the diaphram). press the valve assembly back in and snap the top on again. Turn the water on and your done.
I am including an illustration on how to replace the valve.
