This achievement will come naturally as you play more matches. SMGs, Shotguns and Pistols are the most common weapons to find. A strategy you could use if you want to knock these out quickly is to do a "hot drop" which is jumping out of the plane as soon as you can or jumping into an area that is popular and many players go to. This turns into an arms race to find the nearest weapon which is normally one of the three mentioned above and then hunting/killing the nearest person before you start looting.
Here are all the weapons (with a few bits of info) that apply to this achievement:
SMGs
Micro Uzi - 9mm. Iron sights only. Damage = 26
UMP - 9mm. Accepts all SMG attachments. Damage = 39
Tommy Gun - 45ACP. Iron sights only. Extended mag gives you 50 rounds. Damage = 40
Vector - 45ACP. Only holds 13 rounds unless you get an extended mag. Damage = 34
Shotguns
S12K - Semi automatic shotgun. Takes AR attachments. Damage = 22
S1897 - Pump action shotgun. Can be modified with a choke and bullet loops. Damage = 24
S686 - Double barrel shotgun. Can be modified with a choke and bullet loops. Damage = 24
Sawed Off - Double barrel that takes up the pistol slot and can only be found on Mirimar. Damage = 20
Pistols
P18C - 9mm. Firing mode can be switched to full auto. Damage = 23
P92 - 9mm. Damage = 35
P1911 - 45ACP. Damage = 41
R45 - 45ACP. Can only be found on Mirimar. Damage = 55
R1895 - 7.62mm. Slow fire and reload revolver that packs a punch. Damage = 55