If you are having troubles with this achievement, try to max out your shotgun perk as it will increase the size of its blast. It pops in game as soon as you bag 5 birds with one blast.
(Edit) If you are solely looking for this achievement, other members have suggested you try the Nuevo Leon (Survival) level in Mexico as there are 2 large flocks of birds.
- thanks Brainiac427
for your suggestions.