How to get the different endings in Lenna's Inception.
To get the Sacrifice Ending, kill at least one dungeon boss, and get the Power of Truth by obtaining all Corrupted Memories from the dungeon bosses. Then defeat the Chairman and Delvin and blow into the cartridge.
To get this ending, don't kill any dungeon bosses, but also don't get any of the Corrupted Memories, so that you don't obtain the Power of Truth. Basically, ignore all dungeon bosses to get this ending. Just get the treasure from each dungeon, and leave.
Alternatively, you will get the Catastrophe Ending if you say No when asked to blow the dust out of the cartridge, regardless of whether or not you fulfill the requirements for other endings. However, this might not count towards the achievement.
To get the Perfect Ending, don't kill any of the dungeon bosses. Instead, you first have to get the Power of Compassion, which Daniel will give you after you get the treasure of the fourth dungeon and go outside. Then, use the Power of Compassion on all eight dungeon bosses when they are almost dead. You can tell if the Power of Compassion is usable by assigning it to one of your action buttons. It will start floating in a circle when it is usable. However, the Power of Compassion will only work on a boss if you have the Mystery Item specific to each boss. The Mystery Items will be displayed on the left side of the Key Items screen. One clue to which item you need for each boss is at the beginning of the game, in the school. You can check on each student's desk, and talk to each student, to determine which item each one would like as a gift. The gifts for each boss are random for each random seed, except for the binoculars for Azraflail, since Azraflail is always the first boss.
Since the order of the bosses is always the same for a given World, you could create another save file with the same world name to see the beginning again, and examine each student's desk for the clue to which gift to give them.
If you spared all eight dungeon bosses, then after the battle with Delvin, say Yes when asked if you will blow the dust out. (If you say No, you will get the Catastrophe Ending.) You will still get the Perfect Ending if you kill the Chairman or Delvin, but using the Power of Compassion to spare them will change some details of the ending.
After you blow the dust out of the cartridge, there will be cutscenes, and you will end up in the school. The Game Cartridge will be in your Key Items menu. Use it to go to Level 9.