This is a common problem. The tiny little needle valve in the fill valve mechanism is not sealing completely so an extremely slight amount of water continues to enter the tank even though the valve is closed.
It also sounds like you have had to change the flush valve at some time in the past and the new flush valve standpipe is slightly too tall, otherwise the excess water would be spilling over the top of the standpipe and going into the bowl rather than leaking out the flush handle opening.
You might have some success by adjusting the fill valve float arm to stop the water a bit lower in the tank. (There should be a water level index line printed or embossed on the tank wall) This may be enough to completely close the valve and stop the slow leak. Otherwise you will need to either rebuild your existing ballcock fill valve assembly or replace it.
Unless you have a toilet that has a radically different style of fill valve, i would suggest replacing the fill valve with a Fluidmaster universal fill valve kit, which is available at all hardware stores for about $10. The fluidmaster unit is a complete kit with all necessary parts and very good illustrated directions. This is a simple DIY job that should take about 20 to 30 minutes and all you will need is a good pair of pliers.