To unlock this game mode you must first beat the game with either Dante or Lucia. The Bloody Palace mode is a gauntlet-style survival mode where you will face waves of various enemies and bosses. At the end of each wave you are given 3 portal choices:
A portal that takes you up 1 level
A portal that takes you up 10 levels
A portal that takes you up 100 levels
This is by far the hardest of the Devil May Cry 2 achievements. You will need to pass 9000 levels of The Bloody Palace without dying and without saving. The upside is that you can take portals up 100 levels at a time, you're allowed to use items, and if you're on an odd number of waves you can skip boss fights.
The biggest down side to this is that after wave 1000 enemies and bosses will start using devil trigger (just like in Dante Must Die mode), making them even tougher to kill.
The best advice for this mode is to wait until after you've beaten the game on Dante Must Die mode (so you're accustomed to enemies devil triggering), use Trish (she's the best character all-around), have your character fully upgraded (devil gauge, health, skills, weapons), have a full inventory, and abuse devil trigger whenever you can use it.