Mark the layout. There are building codes that dictate how high a staircase railing should be--generally 30" to 38" for handrails against a wall. Contact your local building and development department to confirm the standards for your area. Using your tape measure, mark two points at equal height--one at the top of the stairs and one at the bottom. Snap a chalk line between those two points. Locate the wall studs with a stud finder and mark their location along the chalk line.
Install the wall brackets. Using a level, mark the positions of the brackets. Ask a friend to help you by holding the brackets in place while you punch a hole in the wall with an awl. With your drill, you can now drill smaller pilot holes and then screw the brackets in place.
Install the railing. Lay the railing onto the brackets and double check that the railing lies snugly in all the brackets. Make sure that the railing is in the correct position, and then mark the bracket holes on the railing. Using the correct drill bit, drill screws through the brackets and into the handrail at the spots you have marked.
Install railing returns. Railing returns are the caps that are secured onto the ends of the railing for decoration and to prevent them from catching on clothing, or otherwise causing damage. Your aluminum railing most likely came with railing returns. Install as directed by the manufacturer.