First make sure the ball float (plastic cup if you have one) moves smoothly.
This problem sounds like the flow isnt stopping soon enough. Screw the stop screw out a bit so that the tank dosn't fill up so much....did that do it?
Now you have to make sure the flush handle is working smoothly and not sticking like it sounds yours is.
No joy?! OK
You need to buy and fit a new ball cock setup from a local B&Q or Wickes.
They are easy to fit and fairly cheap.
Sound like what you have is worn out.
You have to turn the water off, drain the cistern.
You will need a pipewrench type tool because the plastic nut is fairly big.
In general it's an easy job.
Good luck.
Bob the Builder, February 2005