The guides for the original Mafia 2 are still relevant.
This video guide (note mine) shows the method for the achievement.
Key points are:
Stealth takedown the 2 security guards (hold B when behind them) One patrols the building and one sits behind the desk on the first floor.
Getting the key for the safe from the directors office.
Disabling the safe alarm from a room in the basement.
Then opening the safe and stealing the stamps.
If you don't disable the safe alarm, you can open the safe but the alarm will sound and void the achievement. There is a checkpoint just as you initially enter through the window, so you can restart from there.