Deep Rock Galactic

The earliest you can obtain this achievement is when you reach Player Rank 7. Go to the shop on the Spacerig (3 large green panels located between the Abyss Bar and the Mission Select station) and navigate to the mustache tab. Purchase all five for the same character class and the achievement should unlock when you back out of the shop interface.

Edit: As new updates are released and new cosmetics are added, it may be possible to obtain this achievement at Player Rank 6 or earlier. (credit: JohnnyInterfnk)