This achievement can be obtained with either Dante or Lucia, but it should be noted that Lucia's missions are easier and faster. You will unlock this mode after beating hard mode as either character.
To beat this mode you will need to have fully upgraded Your health bar, Devil trigger and Weapons. It would also be in your best interest to carry a fair amount of vita/holy stars, some smell of fears, and gold orbs.
Dante Must Die Stats:
Lesser demons and bosses health: 200%
Lesser demons and bosses damage: 200%
The main difficulty of Dante Must Die mode, however, is that enemies will now activate devil trigger. This means that (when triggered), they will become more aggressive, break your combos, and not be stunned by your attacks. This mode will require significantly more strategy than hard, but it is completely do-able with some perseverance.