How Do I Stop Water From Leaking Into To Toilet Tank?

Often this kind of continual leak is caused by an improperly set float height. Try bending the float bar towards the water in the tank. This will cause the valve to shut off before the tank is quite as full as before, and should stop the trickle (which is usually caused by the overfill pipe draining off excess water from the tank, so it can't overflow).
 You should not have any problems afterward. I've encountered this problem in people's houses, and as I find it annoying, just fixed it for them... (^_^)
 Have a great (and leakless) day!

The chain from the inside handle to the flap could be just a bit to short causing water to run out. See if you can shorten it just by one loop. That's what I had to do.