It could be as simple as checking the stopper to see why water is leaking past it. Mine did that and all it was a dirty stopper. However, without know what kind of a stopper you have, it is hard to say. Flappers are easy to replace, but the plunger types are a little more complicated. If the stopper seat is clean and in good condition, all you may have to do is simply replace the stopper, which will not require removing the cistern. Lift of the lid and take a look at the stopper and see if it is a plunger type, or a flapper.
Feel around on the stopper seat and see if it is not chipped, knicked, or dirty. If it is not, your next step is to check the stopper. Some are removed easily. Hope this helps!