You need to achieve SR-8 to unlock the Warrior armor. As a mediocre player, I got this in about 10 rounds of Big Team Slayer.
Once you have gotten to SR-8...
- Open the Spartan Hub by pressing the Start button
- Select Customize
- Select Spartan Armor and press the A button
- Select Torso
- Select Warrior
- Achievement unlocked
As pointed out by Pagemaster 146:
The Limited Edition comes with armor you can equip before playing any matches. Not everyone will have this obviously but this is an alternate way to get the achievement.