How do I get it to run a script when I level up a certain hero (me)?

From OHRRPGCE-Wiki

There is no built-in script trigger for level-ups, but the effect can easily be simulated in an after battle script.

You can use the hero levelled command in your script to detect whether or not the hero gained a level-up from the most recent battle.

#somewhere above...
include, mygame.hsi

#and, the script
script, after battle, begin
  if(hero levelled(find hero(hero:joe))) then, begin
    #do something
  end
end