When the game starts, the minigames should be on an automatic playlist, which play the entire way through.
The first way you can get this achievement is to let the game go through the entire playlist of games in one go, which will take around an hour. You MAY need to back out at some point and select the credits minigame, which is the man holding the Double Fine logo.
The second way you can do this is by selecting games from the stage area, and keep backing out once the stage has loaded until you have gone through each game.
Either way will bag you the achievement, however I suggest the first method if you want to bag some of the other achievements the game has to offer and so you can also enjoy the game while you're a it. If you just want to get the 400G over and done with, there is no harm using the second method.
The third and fastest way to do this suggested by DummbAsh, is that when the game loads up hit 'RB' until you've gone through each game, this will get you through all the games in around 5 minutes.
LitaOsiris claims that you should go through all channels in one sitting of the game, otherwise it may get reset. This isn't currently confirmed yet although the nature of the other achievements would suggest this to be true.