To unlock this achievement, you must hold the highscore for Double Take when you finish the game. The only problem is, once you've first gotten the highscore once, Izzy will outdo you by two seconds and retake the top spot. Which means you must either be able to do better than your previous best by 3 seconds, or just get the top score when Izzy don't have time to worry about the game!
So don't touch the game until the very end of the story, before you go and do the final puzzle:
*** Spoiler - click to reveal ***During the final day, when there's snow covering the ground outside, and you must go down to the cellar and do the puzzle with the boiler levers.
Right before you go down to do the puzzle, go play the game and get the highscore! Izzy will not have time to get a better score for you, both because it's at the end of the game, and also because she's busy with her own problems.
Just finish the game, and the achievement will unlock!