I havnt done a local high score system but I assume you will need to use a dictionary or array with min 2 columns (names vs score)
you will have to sort the array by score.
then when you have a new score ready
if your new score is greater than the bottom score in the array
push the new score into the array
and sort again
then pop the bottom one
also every time the array is updated then also save it to local.
and load from local every time you start the game.
then use the array text wherever you need to display your high score