At the very beginning of the game, after you talk to the crows and are given your first assignment, head left from the main area. Instead of going through the door that has spawned, keep heading left and up some more flights of stairs. The umbrella will be at the top of the stairs leaning against a lamppost, near a bench.
ImageJust switch your weapon to the umbrella from the menu, and make sure not to use any other melee weapons for the rest of the game. Just picking up the weapons won't nullify the achievements, nor will using or upgrading magic abilities - so you can 100% the game on the same save file as the umbrella run.
Make sure not to swing your sword at the start of the game! Swinging your sword, even in the starting area before you meet any enemies, may prevent the achievement from unlocking. The achievement should pop at the credits after the last boss fight.