In Super Mario Maker you can’t play as Princess Peach, but you can get her outfit by completing a challenge in Story Mode. This challenge is available after you’ve beaten the game and completed the rebuilding of Princess Peach’s castle. Once you’ve completed the rebuild, you can head to the Taskmaster who will have a new challenge for you from the Princess.
The challenge is called Let’s Go Princess Peach and it’s basically a tutorial that will explain what your new gear does. After you’ve completed the game you will get the Hammer power-up which allows you to use a hammer as Builder Mario. Complete this tutorial level and you’ll unlock the Princess Peach dress.
Head back to the Taskmaster for one more job from Peach and you’ll get the March of the Rookie Toads job. This job requires that you lead 10 toads to the finish line of the level and the reward will be the Princess Peach wig.
Once you have the Princess Peach Wig and the Princess Peach Dress you can use them on your Mii character by changing your outfit.
These missions will not be available in the game until after you’ve completed the castle to 100%.