I made some errors and as Kodiak393 pointed out, so I'm fixing this.
One would think that this achievement requires buying enough Heraldry options to completely customize your armor with things like emblems and patterns, but it is much simpler.
You can do this is either by winning Eternal War or Operations missions, but you have a tradeoff either way. Eternal War games are quicker, but heavily dependent upon your teammates. Operations are longer, but a much better chance of successfully completing the mission on the Minimal difficulty setting plus you can level your weapons and character class.This achievement is also one that requires the playing of Operations, rather than Eternal War mode.
Select the class you want to focus on. For me, it was the Heavy. When you go to the Armouring Hall, you'll see Edit Armor at the bottom of the character screen. In the first tab, you can set up a custom Armor set at the bottom. Select one of the slots. You'll see a bunch of options like Helm, Right Gauntlet, Breastplate, etc. In order to get this achievement, you'll need to change each of the slots to an armor option other than the default.
I incorrectly assumed that it was tied to overall levels. Instead, the unlocking of armor pieces is tied to number of Operations successfully completed or to Eternal War matches won. In either case, the number seems to be around 12 successful missions/games for each character class. However, in order to get those armor options, you'll need to level up that character class. For the Heavy, the first chance to get the achievement comes at Level 12, as that is when the Greave slots get new options. Just play the class a bunch, level up, and you'll eventually have the option to change each of the slots.