Singing higher is rarely accomplished by brute force, but this is often the singer's first instinct. The problem is the inborn human ability to yell. Yelling is a survival mechanism, and we are quite good at it. However, the nervous system has a propensity to switch into yelling coordination, and that locks up the singing voice and shuts down the higher range.
To avoid yelling, the singer should access the upper register with a very light vocal mechanism. If the voice is slightly breathy or a type of falsetto, that's fine. Once these light notes are established, the singer can increase vocal cord closure and singing intensity. Gradually building strength is a good idea, to avoid the tensions so often present in poor singing.
Don't Force the Breath
If the coordination of the upper register is weak, excess air will usually cause more problems. Some singers believe they need to keep pushing more and more air to reach the high notes, but this will cause them to either jam up or fall apart. Yes, support and air are critical, but air is a force that must be resisted by the muscles of the cords. Think of someone attempting to lift too much weight at the gym. It is nearly impossible to maintain proper form. Lighten the weight, get the correct form, and then gradually increase the weight—or, in our case, air.
The root problem of high notes is often found slightly lower, in the area many singers refer to as "the break." This is where the voice tends to crack or flip, and singers often add excess tension to avoid the problem. Unfortunately, this robs the singer of the upper range. The secret to avoiding breaks and smoothly blending from chest voice into the higher register (often referred to as head voice) is to use vowel modification. Vowel modification uses the acoustic principle that a more closed or rounded vowel allows the singer to slowly let go of the chest voice and bring in the head voice. Even nonsingers instinctively know they can go higher if they shout "whoo-hoo" rather than "ah-ha."
Use closed vowel sounds such as "oo" or "ee" when singing higher scales. Once you can get into the desired register, slowly open the vowels to "oh" and "uh" while maintaining the resonance of the more closed vowels. This should begin to make the upper notes easier. Another example: When ascending on an "ah" vowel, let it modify to "uh" as you go through the break area. This should eliminate the strain and cracking of a pulled chest voice.
Most singers are familiar with those silly-sounding vocal exercises that have you bubble your lips or trill your tongue while doing a scale. There is a great reason for doing these semi-occluded or partially blocked sounds: The back pressure created by the lips or tongue feeds pressure and energy down to the vocal cords, making it easier for them to phonate and resist air properly. Other handy sounds are the "ng" sound from "hung" or a "zzz" sound. Use these when warming up the voice into the upper register and give the cords a wonderful assist.
Consonants Are Good
Putting a consonant in front of the vowel can provide some of the same benefit as a semi-occluded sound, but in a more musical way. Hard consonants such as "g" help with cord closure while voiced consonants like "m" and "n" keep the cords phonating the whole time, giving them a healthy workout without the stress of having to do all the work. "Mum" and "no" are good exercises for this.
Repeat Repeat Repeat
Repetition of problem notes on a "friendly" exercise can be very helpful. A friendly exercise or sound is one that is easy for the singer. For every singer, certain vowel sounds tend to be easier than others (yours may be "mum"). Find these friendly sounds and do repetitions of them on the problem notes. This gives the nervous system a chance to experience the necessary vocal balance and ingrain it into muscle memory.
Start at the Top
Starting exercises from the top note and working down can often help lighten an overheavy voice. This breaks the cycle of continually having to get out of the chest voice to access the upper register. Starting from the top also forces the singer to lighten the voice, meaning less squeezing of the vocal cords, raised larynx, etc.
When working on the problem notes of a song, replace the text with a vocal exercise. Try "noo" on the high notes, gradually adding the text back in. Again, you can also use vowel modification to find appropriate substitutions for problem words. For example, singing the word "that" in the upper register is going to be much easier if it's pronounced "thet."