This achievement sounds a little deceiving; you don't actually need a full set of vanity-specific clothing to earn it, any armor or clothing will suffice.
The easiest way to get this achievement if you're just starting is to begin mining. Near the surface will be an abundance of copper ore. Smelt it down with a furnace, and turn it into armor. You will need a total of 60 copper bars (180 copper ore) to make a full set. Rather than equipping the armor to their regular place, just move it over to the right, the vanity slots, and your achievement will pop.
If you're already started & have armor, even mismatched armor pieces, just move them over to the vanity slots. Once they're all there, it should pop up.
Hope this helps!