Fix the Flap.
You have diagnosed the source of the problem with your running toilet and it is the flapper. Or it’s the float. No matter, it has to be fixed before the sound of a mounting water bill drives you nuts. Turn the water off at the valve which is located a few inches above the floor under the tank and flush to mostly empty the tank. You will have to turn the water back on so you can flush to check each attempted repair.
Let’s start with the flapper.
If the water draining from the tank is seeping rather than running it may be that lime or calcium deposits are keeping the flapper or the tank ball from fitting firmly in the outlet pipe. Take a piece of emery cloth or steel wool and burnish the rim of the drain and wipe off the flapper or ball with white distilled vinegar and steel wool. You can probably do this with the part still in place.
If the water is running out faster the ball/flapper may have outlived its usefulness. The flapper especially is prone to rot or to become rigid with age. Those drop-in cleaning tablets that turn the water blue are very hard on rubber and plastic too and many plumbers advise against their use.
The flapper may be getting hung up in the chain that lifts it or the chain may be getting hung up on something. Removing a few links from the chain may solve that problem. Conversely, make sure there is enough slack in the chain for the flapper to relax into position. If the closing mechanism is a tank ball on a wire, check the wire to make sure it is directly above the water outlet and is straight. Straighten the wire if possible; if not it will have to be replaced.
If a new flapper or tank ball must be installed, remove the old one – on both toilets in my house this is as simple as unsnapping the arms of the flapper from either side of the valve housing; tank balls unscrew – and take it with you to the hardware store to make sure you get the right replacement part. To install merely reverse the procedure for removing it. That’s it!
Stopping the water from coming into the tank is a bit more complicated than stopping it from leaking out. We will tackle this next.