A lot of it seems quite informal, so I'm not sure exactly how you want it to sound. I've tried to stick to proper grammar, so let me know if that's not what you want.
Hello, my name is.... but most people call me.... I was born on March 25, 1993. I live in Chiangmai, Northern Thailand, with my big family. I live with my mother and other relatives (grandmom, uncle, aunt ...).
My father not living with us, because he is busy with his job. He is a politician, so me and my mother understand him. I meet my father every weekend, or 2 -3 times a week. We go out for dinner almost every week.
I don't have any brothers or sisters so that's a bit boring sometimes but I am not sad about it because I'm not that kind of person. I'm an active, easy-going and talkative person and I love to talk with new people.
I am self-confident. That's why i think i different from other students, but sometimes i have too much confidence and it's not good for me. I love sports like basketball, badminton and volleyball. I like to try new things (except if it too dangerous, then I won't try it, hehe).
In Thailand i can't ski because we don't have snow here. That makes me a little sad. I wish it would snow one day in the future. Oh and not just sport, I also love music. I can say music is my life.
I can play piano, guitar (just a little), and some Thai music. Well, actually i can play more than I have wrote, but I haven't learned the entire song yet. [is this what you meant?] I mostly love to listen to the music. I like R&B and some hiphop songs.
I don't like sad songs though, because it's make me feel blue. Wow I think I've written too much about music.
Back to myself, I am interested in other languages, including Spanish, Polish, German, as well as the cultures and social lives in these countries. In my free time I like to play on the computer, sleep, listen to music, and read about philosophy. When I was young my mom always bought me philosophy books, so i familiar with lots of philosophy. I don't like to read a comic books like other teenagers.
As i told you before, I'm an easy-going person, but i'm not going to tell you that i'm perfect. Nobody's perfect after all.
Sometimes, i'm a lazy person like other teenages, but I'm still responsible when it comes to my work.
My ambition is be a rich business woman and the lawyer at the same time,
so I need to study hard and be more mature. I hope that my life will be better and better, and I hope that my stay in America will be a rewarding experience with no trouble. Finally, I hope all my dreams will come true and I will be very happy if I succeed in my studies and in my life.
I've put corrections in red, and stylistic suggestion in blue.
If you're going to send a letter, then I would keep it more formal. If you are going to email them, then feel free to be a bit less formal, and in the bits where you say things like ( * Big Smile on face) or (*crying on her face with a wistful eyes), use the smileys instead.
Not a teacher.
Student or Learner