This achievement can be obtained as soon as you reach level 10.
EDIT: From what many people have been telling me, if you preordered this game or have the Ancestor DLC then you may be able to get this sooner so check to see if you have anything unlocked.
First, select Multiplayer.
Next, at the Multiplayer menu, hit LB until you see a list and one of the options is Characters so select that.
You will see a list of all the different characters you can choose to play as in a Multiplayer match, select any character you want.
The first thing you can change is at level 10 and it's the Main Weapon that your guy uses, so select that.
You will only have one different weapon to change it to and it costs only 25 points and is available when you reach level 10.
Change the weapon then save it and the achievement will unlock immediately.
Could a step-by-step guide get any easier?