How are traits leveled up for the Competent, Specialist, and Expert achievements?
According to the manual
All traits start at level 1 and may be improved up to level 5 by successfully completing missions and scenarios using these traits. Improving a trait either increases a positive effect granted by the trait or decreases a negative effect 10 imposed by the trait. Trait level is saved automatically. Traits can never go down a level.
So it sounds like the key is to use each trait during a successful game.