What is the correct way to seal off a fireplace chimney?
Whatever area you live in I would suggest A minimum of capping the top, then blocking the bottom.
Capping the top prevents moisture like rain from causing problems below. Blocking at the bottom helps future residents from trying to use the fireplace.
I have seen perfectly good chimneys capped because the fireplace was so inefficient, then a new owner DIY a insert and caused thousands in smoke damage.
My current home had a 1930 fireplace with a gorgeous river rock chimney, it was not lined so at first I capped it then removed the chimney.
If I would have left it inplace for the Look I would have blocked it also at the bottom.  Just in case...