Shut off the water supply to the toilet. Remove the tank lid and flush the toilet, draining as much water from the tank as possible. Absorb the additional water in the tank with the towel. Disconnect the thin refill tube from the tank's overflow tube. The refill tube is the thin tube with one end connected to the fill valve and the other end connected to the top rim of the overflow tube, the pipe rising from the tank bottom. The refill tube is normally connected to the overflow tube with a small clip.
Place the bucket beneath the tank where the water supply hose is connected, to catch the falling water once you disconnect the supply hose. Loosen the connecting nut with the adjustable wrench and unscrew the nut by hand. Pull the hose free.
Lift the old fill valve from the tank and lay it on the floor next to the new fill valve. You can adjust the height of fill valves, so adjust the height of the new fill valve to match the old one. Place the new fill valve's rubber washer over the bottom of the valve and slide up as far as possible. Place the new valve into the tank with the valve's bottom shaft through the hole in the tank bottom. Make certain that the valve washer is seated flat on the tank bottom.
Hold the new fill valve in place while sliding a lock nut onto the valve bottom beneath the tank. Hand tighten the nut, but be sure not to over tighten and crack the tank. Connect the fill valve's refill tube to the overflow tube. If necessary, adjust the valve height so that the point where the refill tube is connected to the valve is about an inch above the top of the overflow tube.
Connect the supply hose to the bottom of the new refill tube. Hand tighten the hose's connecting nut and. Do not make it too tight. Turn on the water to the toilet and allow the tank to fill with water. Check beneath the tank for leaks. Replace the tank lid.