This is not a cumulative achievement. It will unlock when one of your characters gets $5000 saved up all at once. This includes characters you're not using, who gain money (and experience) as you complete quests.
If you were to play one of the girls only, the other girl would have $5000 about halfway through a New Game+, and the achievement would unlock at that point without having to switch over. This is the best way to go about it if you're a solo player, as you really need to be buying all the food and skills you can on your active character, and your other girl will be too weak to grind with.
If you're trying to grind this out on drops alone, there's no special way to do this other than just going from screen to screen killing everyone. This is extremely slow, however. The coin purse accessory from the shop in front of the Noize concert will give you a 5% chance at double money drops, but it hardly speeds it up. I strongly suggest just completing major quests to get the money fastest.
Also, before you begin your NG+, you should probably fight the secret boss since that counts as a quest and gives a huge amount of experience and money. The secret boss is accessed from the same room as the final boss by equipping an accessory you get from destroying all the statues, and another accessory you get from simply beating the game. Hasabe Charm and Mami Charm.