You can buy new clothing directly from the shop that appears when you’re applying for Career and Runway Challenges. When you tap on any item category, for example “Shirts”, you’ll see your closet divided into the “Shop” and “Owned” tabs.  
You can buy new clothes by tapping on a piece of clothing in the “Shop” category and tapping the “Buy” button. Once bought, the clothing can be accessed from the “Owned” category.  
 
Another way you can buy new clothes is to simply dress up your model in any clothing you like while you’re dressing up for a Challenge, and after tapping on the “Review” button, you’ll see all the clothing you used, and if you wore anything you don’t own, a “Buy” button will appear.  

Some clothing is only obtainable through clothing collections. You can access collections once you reach level 6, from the bottom menu tab with the clothing hanger icon. You can get collection clothing from Fashion Card packs that you can buy with Star Coins. You can get more Star Coins by replaying Career challenges or exchanging them for diamonds.