Begin by attaching the shelf boards to the side supports as shown above. Choose exterior decking screws installed correctly (see packaging) to prevent water damage. If you have a you could also fasten from the underside and completely hide all screw holes. Take care to ensure that both (Yes, make two) shelves are square. Glue is advisable.
Back supports are necessary to keep the project square. You could also do simple cross bracing for the bottom if you prefer that look. The most important part is that you 1x3s supports are flush to the top of the shelf tops, and you screw into legs, shelf boards and shelf supports.