Also make sure you have at least a little slack in the chain. I had yet another recurrence last week of a chain issue. We once again had a toilet "leaking" in our home and the fill valve just very slowly running. Upon initial inspection, everything seemed in order, but I knew to check the chain and, sure enough, I noticed that the chain was tight, rather than having even a bit of slack like I had previously set it. I realized that the flush handle had rotated just enough to be pulling the flapper ever-so-slightly off the seat. Tightened the flush handle in its new position (because the rotation seemed by design to be able to go no further; remember, they are reverse-threaded) and added two links to the chain, and -- voila! -- problem solved.
Given that you replaced the flapper, a chain length issue might be indicated. (Also, make sure the converse isn't true, that the chain isn't bunching under the lip of the flapper because it's too long.) Recognize, too, that sometimes putting the lid back on can alter the trip lever position. Probably not indicated in your toilet, but worth a mention.
Finally, and maybe this is obvious, but make sure that the water level you set is below the top of the overflow riser and you're not getting running over the top of the riser. From your description, again it doesn't sound like the issue, but worth a mention.
Nine times out of ten, it's what Terry mentions -- a siphon caused by the refill tube being just stuck down the overflow riser rather than being daylighted on a clip above it. So that's obviously where to start.
Come back if none of this works. And if none of this works, also tell us what kit you used so we can think of other possibilities.
Good luck!