Choose your materials for the flower box. For example, you may use a plastic container -- these retain water more than materials like wood, which absorbs water -- so that you don't necessarily have to water your plants as much.
Punch or drill holes in the bottom of the container to drain excess water.
Attach railing brackets to the container according to the instructions on the package. Railing brackets attach to your container, then hang on your balcony's railing.
Fill your flower box with soil and plant your flowers. Water the flowers according to the requirements of the type of flower -- some need more water than others.