You gain experience by "blowing shit up" simply try your best to kill anything that moves without your pilot dying then when you run out of ammo hold down the SELECT button to switch to a different plane.
By the end of the 5th mission on the first US campaign you should have reached 255 EXP, the achievement will unlock at the mission summary screen
To make it easier on yourself, when editing the plan loadouts, change the fighter guns to 6x machine gun turrents, it simply shreds everything.