Edited now, as it appears the fill valve doesnt shut off, so the internal overflow is running. New fill valve. The old one can be adjusted slightly, but IME, if they have staretd leaking, they need changing. Certainly easier than changing the flush valve, below.
Your flush valve is either damaged, or has dirt on its seal.
There are a number of different types, and, at least two sizes of outlet available, as well as different volumes in the older ones.
Changing it can be anything from 2 mintues to several hours.
Some just unclip, then you clip in the new section, leaving the bit that is in the bottom of the cistern in place. 2 minutes,and you dont even have to turn off the water.
Others will need the cistern removing. Firstly the bolts can seize and rust.Once they are off, and youve got the cistern off, it can be a right pain to get everything water tight. Scrupulous cleanliness and sealant paste work here, as well as tightening the correct amount. Then the rubber ring that seals between the cistern and pan can degrade over time, so may need replacing. Again, they can be a right pain to seal properly, but experience helps there.