First, be sure that when the water stops filling, the level is about 1" below the top of the overflow tube inside the toilet and not so high that it is slowly leaking into the overflow. That would require an adjustment on the fill valve.
If that is okay, find a comparable flapper replacement for your brand of toilet at a hardware store or big-box store (Home Depot, Lowes, etc.).
At this point, it's a process of elimination. While at the store for the flapper. Buy a few dye tabs. They will be in the toilet repair section, too.
Replace the flapper, let the tank fill and put in the dye tabs. Let the tabs dissolve and do not flush. This may take some time and, as always, read directions provided.
If the flapper was the correct replacement and you can see dye running into the toilet bowl, you may have a leaking flush valve.