Just get one kill with each weapon in the game. There are 10 weapons total, and if you're bad at certain weapons, you'll eventually get a kill with one by blind dumb luck. Besides your starting weapons, the rest of the weapons are spawned on the map that you must get. Sometimes other players are doing the same, so get there first, or try to take turns.
Make sure that you actually get the kill.. a person downing an enemy and you shooting them will only count as an assist. You can down an enemy with one weapon and kill them with another if you wish.
Press [LB] (Tac/Com) while in a multiplayer match and it will show the weapon spawns on the map.