After you purchase items they will appear in the inventory where those things are customized. First select Options from the main menu. Then Appearance or Avatar. Items and Tokens are not shared inventory; make sure you are signed in under the user name you wish to buy/equip tokens and items for.
To customize Appearance select the category you wish to manipulate and equip your items or purchase more. The ones you own appear in the top portion. Highlight the one you want on this profile.
When you select Avatar from the Options menu your customization menu looks like this:
You can select your avatar from your inventory here. There is also a Revert option to reset any avatar manipulation you have
Use the slide bar at the bottom to reveal more of your inventory.