To unlock this costume and achievement, you will need to visit the Orcrest Public Dungeon in Elsweyr, kill mobs and find 10 Plague-Drenched Fabrics. The items drop from any mobs in the dungeon, not just bosses.

The Plague-Drenched Fabrics are not character bound and can be purchased from guild vendors if you are unwilling to spend the time grinding in the dungeon. 

You can track your progress by visiting your journal and finding the relevant section. Playstation and Xbox players can also unlock platform-specific Trophies and Achievements by completing some of these tasks.

Peryite Skeevemaster Costume: Male and Female