You'll need to collect at least three of the below hats and then equip them on any given mission to unlock this achievement. Please note that you'll need to equip three different hats, you cannot use the same hat twice.
Name: Crown
Owner: Scrapper Swab
Location: Cargo Barge
Name: Royal Crown
Owner: Royalist Queen (The Red Queen)
Location: An Audience With the Queen.
Name: Royal Wig
Shop: Le Chapeau de Licorne
Water Cost: 4000
Rep Cost: ---
Name: War Crown
Shop: Le Chapeau de Licorne
Water Cost: 7000
Rep Cost: ---
Credit to Woudo from the Steam Community for the information.