Players can cycle the skin they want to use by pressing the Pause button on their controller (or 1 NUM key with PC keyboard) in the character select screen. From there, a "Select Costume" menu will pop up and you can confirm which skin you wish to equip.
While the appearance of the skin is not previewed on the screen like newer MK titles such as MKX or MK11, you will see the skin as soon as you jump into any fight.