You need to hit as many targets as you can before the timer expires. To get a 5-gear rating, it will be easier if you pass the challenge the first time around enough to earn the Precision Aiming mod. Enable that mod then try the challenge again and enter precision aiming by pressing L3. This allows you to shoot more accurately.
Unlock by: Reaching the province after completing “Connect the Dots” story mission.
Zoom in to have steady shots. Aim for the center of the target and keep shooting. Since you’re using a machine gun, it’s easy to miss and reset your multiplier but it’s also easy to build it. Just control your shots and avoid missing or hitting the blue X boards.