There are 6 weapons in the game, 3 are picked up from enemies, the other 3 are given to you as part of the storyline. The earliest you can get this achievement is during Breaking Camp (chapter 2 part IV) after you have been given the last gun.
I'm not going to list where exactly the story guns are as that would involve spoilers, but suffice to say you can't progress through the story without them.
Liberator
The human rifle you are handed by a survivor at the beginning of the game, unmissable as part of the storyline
Opressor
The basic alien rifle you can pick up from most dead grunts
Disintegrator
The shotgun/grenade launcher gun that you can get from larger enemies
Reclaimer
The sniper rifle found on enemies - you first spot these during mission Prophesied One and they appear throughout the game, watch out for the laser sights targeting you
Magnetar
The gravity gun which is found during the mission Prison Escape as part of the storyline
Hypercoil
The energy gun you are given as part of the mission Breaking Camp as part of the storyline
Simply kill an enemy with each of these weapons to unlock the achievement. You can only carry two guns at one time, but you can change your loaded in the inventory screen or pick up different guns in the field.