When attempting to break open doors, be sure to press forward the Right Stick once the bar hits the very center. Simple as that.
If the bar moves too quickly for you, you can just not complete the command and wait it out until you can time it well, otherwise you'll have to keep hitting the door and attempt the achievement on another door.
No worries, folks! There are plenty of doors in the game to try this at another time. Otherwise, you could just start up a new game.
During my first playthrough (as Logan), I ran into a break-in door almost immediately. Then a second while on-way to your first quest's goal.
As a side note, I'd like to add that counting to four does help as well. By that point, you'd have picked up the rhythm to hit the center.
"When you reach the Lifeguard Tower, you are given a Quest to break in to the Medical Room. If you don't break the door in on the first try, you can restart checkpoint and it will put you right in front of the door again."
- Callum x360a
Thanks for that tip, Callum.