In this mission, you have plan A and plan B. If you do plan A, you can end this mission in less than 10 minutes! So, firstly sneak to the alarm boxes. The guards only notice you if you too near and in their flashlight range. Just stand and run if no guards see you.
The position of the alarm boxes depends on your luck. If you're lucky, all three can be at the same floor. :D
It should be easy when you practice sneaking many times.
When all alarm boxes has been hacked. Now you must sneak again to the vault to enter the codes. Do not blow your cover now, if not the codes will be not usable.
Yeah, the vault opening! If not, you will have to get another code. :(
Take all the diamonds fast! The longer you stay, more police will come.