Should be really simple - use the Webstorage object to save a local key, which contains the high score. Then, at the end of the game, just compare its value with the current high score, and if the current score is greater, update the local key's value.
<font size="2">(If you're developing another, yet ANOTHER clone of that stupid game, i have no respect for you; we have enough of those already, be original, try thinking for yourself, goddamnit)</font>