In each shop there are an assortment of items to buy. These items are all part of a set which contains 3 pieces to it; a hat, a costume and a pet. Some items however have to be won through attaining certain targets in a mini game - 16+ rounds in Boxing for instance.
Once you obtain all three items for one outfit, the achievement will pop.
The easiest full outfit to compile is in the shop called Barkers Best. Here you can purchase the Monkey (Monkey Barker) the costume (Barkers Best) and the hat (Barkers Bowler). In total, these will cost you 750 tickets but will also earn you the "Monkey Barker", "Stylin" and "Tip O' the Cap" achievements.