After beating the game, go to the final level, grab the veil there, and then go to the far corner of the level away from the guard and use it 30 times. The fastest way to do this is to trigger the checkpoint, trigger your veil, trigger the checkpoint again, then reset the timer. This means you only have to wait for the veil cooldown and not through the whole invisibility.
After you've gotten ~30 uses, go to Another Face, grab the costume there, and use the costume. I didn't get this achievement until I did that, even though I had already gotten enough uses of the invisibility item.