While in the Hero Select screen, after you have locked in your Hero for the match, the option to select the Skin you want to use will pop up (if you don't have any skins for the selected Hero there will be no pop up). During Draft Mode, if you Swap heroes with another player you will again get the skin pop up for the new hero you have changed to.
Tap the skin you want to use and it will be highlighted. You will see the splash art for the skin you chose while loading and then you will be using the selected skin in your match. Enjoy!
If you want to change your skin, or make sure you have a skin selected, tap on your hero again and the option for skins will show under the WP/CP/Support options, select it and then select the skin you want again. This can be done in Draft Mode after your swap heroes as well.