In the tutor's email, you should explain in detail that you are going to apply for his postgraduate, and then introduce yourself. You can attach your report card and some positive materials, and don't write anything to brag about. It is better to have some hardware such as scanned copies of awards, your papers (excerpts) and so on. It is also necessary to profoundly explain why you chose this major, be true and don't talk big, write your understanding of this major, and why you chose to apply for a teacher across schools.
Finally, write down your vision and plan for the future, and let your tutor know what you want and what kind of life you want in the future.
It is necessary to show respect to the tutor, but it should not be excessive. If you go too far, you will feel hypocritical and frivolous, and you will pay attention to sinking down when you are in graduate school. Therefore, hypocrisy and frivolity are taboos.