Sounds like the "ballcock" in your water tank is either sticking, or the washer inside it is damaged.
To stop it overflowing, you will probably have to have it replaced.
Do NOT tie the ball in the up position, as the tank will empty and start to cause other problems.
If you feel confident. turn the water off at the mains, check that it's off, by pushing the ball down and listening. (You'll hear and see that no water comes out of the cock)
You can then, if it's a plastic one, undo the unit to check if the rubber disk (Washer) is worn.(Sometimes you can just turn it over and use the other side )
If its a brass one, then you should probably get a bloke in....But don't let him stitch you up....It shouldn't take more than 45 mins to change.... And if you get a replacement from B&Q...even quicker.
Best of luck.
Chumley
November 2009