First if plastic type arm at the water inlet end is an adjustable plastic bolt with locking nut on it. Try adjusting it turning it clock wise to get it closer to the valve pin sticking out of the valve housing. Allow cistern to fill if it still carries on adjust further. If it then stops job done if no further adjustment left then follow instructions below.
Your water inlet valve washer requires replacing by the sound of it. Plastic arm type with fairly large Knurled nut at water inlet end at this end you will see a plastic pin/plunger sticking out which meets the adjustable plastic bolt on the end of the arm this is what pushes against the washer to turn the water off.
Turn water off to cistern and remove the holding pin on the arm at the water valve end. Remove the large knurled nut and pin/washer plunger note which way round the diaphragm washer is and remove the large rubber diaphragm washer replace with new one obtainable at most plumbing oujtlets and DIY stores. Re-assemble in reverse order. check flow in and if still filling too much adjust the plastic bolt on the arm to cut the water off sooner turn in against the valve pin
Some plumbers will charge £100 for a job that takes about 2hrs most allowing for problems of which i cant see any.
If it's the old brass type with a brass bar/float arm and brass valve then email me at robertwhiting76@hotmail.com because adding more now could cause the page to ignite.
I'ts and easy job and get stuck in that's how I learnt.
Nobby
April 2008