This is a fairly organic achievement to unlock, and there's a good chance that you'll get it with simply putting in enough time playing Dogfight matches. The achievement requires you to shoot down four of each Starfighter, for a total of thirty-two kills. The Imperials have the TIE Interceptor, TIE Fighter, TIE Bomber, and the TIE Reaper. The New Republic has the A-Wing, X-Wing, Y-Wing, and U-Wing. You'll need four kills on each one.
If you're struggling to unlock the achievement, focus on destroying TIE Reapers and U-Wings. These are the least used starfighters in Dogfight as they're support craft, and the other starfighters are more appealing to most players. The U-Wings and Reapers are a little beefier on health and usually have decent missile jamming, so push more power to your laser cannons for that quicker kill. From personal experience, the U-Wing was my last starfighter to to shoot down for the achievement.