This achievement is quite easy, but takes a lot of time (upwards of 300 - 400 hours).
Simply play each class enough to get them to Gold 1, which will require 7 promotions.
I personally alternate each time - max one class, then get their promotion assignment while playing as another, max that one, etc.
It's good to avoid playing a maxed out class if you're going for this achievement, as wasting XP makes the whole process slower.
It's also good to prioritize double XP and double warning missions (there will always be 1 double warning mission per mission cycle) and play on a higher difficulty (hazard 4 or 5), that way you can get more XP per mission.