Warm up your voice. Warm ups can include humming your favorite song or singing something like "Row, Row, Row Your Boat". Another great exercise for range increase includes humming your lowest note, then moving up to your highest and back down like a siren. Do it a few times, and you should already notice a difference. It is always best to warm up the middle voice before moving to the high and low extremes.
Breathe correctly. You should be breathing with your diaphragm, in other words, when you inhale, your stomach should rise first followed by your chest. Stand up straight, and try singing some quick "ha, ha" to feel quick the quick puffs of air that come right from the lower parts of your belly. This is called "support" of the voice.
Start in the middle of your range and sing higher and higher. Keep singing along the notes scaling up and down. Never strain your voice. Your throat should not hurt. Drink enough water to hydrate your vocal cords.
Think about the whole phrase that includes the high note. You must support your voice continuously from the beginning, connecting the high note to the notes before it.
Try singing the note while pretending to throw a frisbee with energy.
Remember that "high" notes are not actually physically in a higher place. They originate in the same location as the rest of the notes. Avoid looking up or tilting your head upwards, and try thinking of the note as farther away from you, rather than somewhere up in the air.
Find the right vowel modification. Each voice has certain vowels that work best when one sings above the staff. You should experiment to determine which vowels work best for you. Once you have an idea of which vowel works best, modify (gradually) towards that vowel as you ascend the scale. (Don't be afraid to use mixed vowels like singing the vowel in "head" while rounding your lips like an "oh".)
Yawn on those higher notes. When rehearsing, never hesitate to sing the word "yawn" for one of those notes in your upper range. The position of the mouth, to say that word, is the exact-perfect place to start when you're singing that note. It sets up the back of your mouth/throat perfectly.
Repeat. Throughout a period of time until you get the desired results.
Try singing your highest note and holding it out, then try the next one.
Take your time. High notes are not that easy so do it bit by bit.