During chapter 2 you will have to meet Kaito at a location called 'Koi Bridge', after a battle against a few standard thugs with weapons you will be told to go downstairs to the secret casino.
Go behind the counter and unlock the door, you will be given 300 chips and told to waste some time before the story can proceed.
Create a save here, you will need to get yourself up to 600 or above. You can choose poker if you fancy your luck but it is harder as you are up against multiple CPU's rather than just the dealer in Blackjack.
For Blackjack, win your first hand and then increase your bet on the second. Repeat until you win both hands consecutively. If you fail, cancel to menu and reload the last save.
You must exit the minigame for the achievement to pop, do not risk continuing to play after you have passed 600 - you keep your chips and can use them to play later.