To obtain this Achievement you need to battle 265 enemies to get their names added to your enemy book. This will not be obtainable through the main Quest, you will also need to complete the side quests as well as the Bonus dungeon to complete this. The bonus Dungeon will require you to be a high level to defeat the enemies.
You can do it in your first play-through although it would be easier in a 2nd with 2x or 3x EXP. I was able to beat it with Lvl 98 characters however the Final boss is Lvl 124 and is resistant to everything. I went in with Lvl 60 characters and had to grind up to 98. I failed several times before I was able to succeed.