Once you start a Foundry Character, the only way to customize it, is to create a project starting at ESD. Run the Project, and you will be able to 'play' the character as normal in a closed instance. You can run all the normal missions, visit the tailor etc. from INSIDE the foundry.
Your foundry character is totally separate from your normal characters and can only be used / played / leveled in a foundry mission you create and alone in the universe (basically single player).