Remove the staircase railing if possible, using the appropriate screwdriver. Although this step isn't completely necessary, it does help give you some extra space.
Drape a drop cloth over the entirety of your stairs. Tape the drop cloth to the walls along your staircase using painter's tape, in order to keep it securely in place. Get your paint, your paint tray, and your extended roller out as well.
Attach ladder levelers to the bottom of your ladder. These are like "extra legs," which can be adjusted individually. This allows you to extend one leg further than the other.
Attach ladder pads to the end of your ladder that is leaning up against the wall. These pads protect your walls from getting scratched.
Set your ladder up on the stairs (on the top stair), and extend one of your ladder levelers so that the ladder is level and secure. One leg (the shorter one) should be up on the higher stair, while the other leg (the extended one) rests on the lower stair.
Climb up the ladder, and remove any cobwebs. Attach a pot hook to your ladder so that you can hang your paint can from it. This allows you easy access to the paint, instead of having to climb down the ladder after each time your brush gets dry.
Cut in the walls with a 2-inch angled paint brush. This is a technique that involves painting the edge of the walls, without hitting the joint where it meets the ceiling. Use your angled brush to carefully create a line of paint along the wall/ceiling border. Think of this as creating a buffer, much like when coloring in a coloring book. By making a border along the edge, you're more free to color "inside the lines" without the high risk of making mistakes.
Paint the walls and ceiling, using your extendable paint roller. Start with the ceiling, and the tops of the walls, then work your way down, moving the ladder down the steps as necessary. Adjust the ladder leveler when appropriate. Let one coat dry fully before adding a second coat (if necessary).