One common part to check is the flat circular washer that seals the inlet water off. It is activated when the float lifts, and leaking can mean that washer is on its way out.
As a short term fix you can often adjust the float valve to work normally. Between the arm and the float is a piece of metal wire. Where the float attaches to the metal wire there should be a little metal clip you can squeeze. When you squeeze it you should be able to move the float up and down the wire. Move the float further down the wire. This will make the water shut off prior to hitting the overflow vents. - MitchellM
If for some reason you can't adjust the float enough you can replace the whole float. Bunnings sells a Fix-a-Loo Hush Flush bottom entry cistern inlet valve for about $20. You would still need to adjust the float hieght on the new one. - JDE
If fixing the float position solves the problem, then leave it alone. Toilet cisterns can be a never ending source of annoyance once you fiddle with them. It's sometimes easier to buy a new cistern & lid kit, and it's shiny and clean too! - SalBay
The most common reason for water leaking into the bowl is that your seating washer needs to be replaced. The flush button mechanism actions the seating washer. A little test you can do to find out if it is the seating washer is to turn that tap off. If the water is still running into the bowl after 5 minutes, it is the seating washer. If the leaking stops almost immediately when you turn the tap off, it's the washer in the inlet valve. - MitchellMc
Make sure to turn your water off before releasing and removing the inlet hose attached to the inlet filling valve. Take "extreme" care as the plastic may be brittle with age. Any accidental damage to the holding cradle will mean that you will have to get a new dual flush mechanism. Here is a link to a video from Caroma. Please note that although the person doing it removed it easily it is because the unit is still new. - redracer01