Question

Не могли бы вы проверить письмо? исправить ошибки или может добавить что?
мне очень срочно
Noyabrsk,
Russia
3 April 2016
Dear Kate,
I was glad to get your letter. It was great to hear from you. Sorry I haven’t written for so long but I was busy in my school.
I'm going to pass exams on society and history, because I`m going to enter the Faculty of Law. I hope I successfully pass my exams.
Who you want to become in the future? And when are you going to take exam? What university have you chosen?
Sorry, I must finish my letter now, because I have to revise for my tomorrow exam. Hope to hear from you soon!
Love, Alina.

Answer 1

......exam in society....
I hope I 'll pass my exams succedsfully.
What are you going to be(to become)
in future?(так лучше спрашивать о будущей профессии)
What do you want to be(to become)?
можно так. но не who.
When are you going to take (to pass)your final exams?
(это же выпускные экзамены?значит final).

Answer 2

Noyabrsk,
Russia
3 April 2016
Dear Kate,
I was glad to get your letter. It feels good to hear from you after such a long time. Sorry, I didn't reply sooner but I've been busy with school.I have my finals in a couple of months, so I study a lot these days.
I will take social studies and history exams because I`m going to enter a law school next year. I hope I will pass them.
What about you? What do you want to become in the future? And when are you going to take your final exams? What university have you chosen for yourself? 
I'm sorry, but I must finish my letter here because I have to study a bit for my tomorrow test.  Hope to hear from you soon! 
Take care.
Love, Alina.

Ошибок вроде немного, просто постаралась сделать менее формальным