Begin by singing a scale starting on F above middle C. F above middle C is the first white key before the set of three black keys on the keyboard.
Continue playing each key and matching the pitch of the piano. Move from white to black keys so that you are singing chromatically. Sing until you hit the highest note you are able to sing comfortably.
Determine the pitch that is five half steps or keys below your highest pitch. If your highest pitch is C, five half steps below will be G.
Start on the pitch obtained from Step 3. Sing the following pattern in half steps, from the starting pitch on the sound "G": up, up, down, down. If your pitch was G, you would sing G, G#, A, G# and G.
Continue to sing this pattern up a half step higher each time. If you started on G the first time, move to G# and then A to repeat the pattern from Step 4.
Practice this technique daily to gradually increase your high range.