This boss fight is the most unique one in the game, and probably the easiest.
Like all remixed boss fights, you play as a set character (so you don't need to wait until you level up more, and you can't build up your character to be stronger against the boss) and if you die, you're put right back outside the boss door with no penalty (you don't lose any money or progress, and can attempt again right away as many times as you want). Now, when you return to the boss door in the Dungeon and enter it, you'll be able to fight the remixed version, Astrodotus and Herodotus. This will open the boss door for Herodotus in the Dungeon permanently (meaning on all subsequent runs, so you don't have to worry about beating it on this next run). Once you die and are back outside, talk to Charon to give him the Obol. Just ensure that once you get the Obol, you don't accidentally replace it with a different item from a different Shrine. This is complete RNG, both with finding a Shrine and getting the Obol, but my experience was that the game was pretty good about granting you the Obol when you were after it. Once done, you now need to find a Shrine Statue anywhere (on the current run or any subsequent run) and hope that the item you get from the Shrine is Herodotus's Obol. To access the remixed version of this boss, first you need to beat the standard version. This achievement is for beating the remixed version of Herodotus. Once you defeat the final boss, you'll be forced to start New Game+ immediately, and you will have to meet the requirements below all over again to get this achievement.
ROGUE LEGACY CHARON OBOL FREE
The reason I have flagged this achievement as missable is because there is no free roam or any way to access your playthrough again after you beat the final boss.