(the practice these days is to omit commas in the address and after the greeting)
I am writing to apply for a course at the Effective Learning School, which I saw advertised in an English language magazine. (or : as currently advertised in Education Weekly.
I would like to improve my English because of getting a better job - better: in order to seek better employment. even better: in order to advance my career prospects. I have been studying English for three years, and my written English is very good (according to my English teacher). I scored 85% in my last grammar test, but I lack confidence in my ability to speak fluently.
I would be available to start the cource at the beginning of July or August.
(this is not clear. Do you mean either at the beginning of July, or you could start at the beginning of August? Or, at the beginning of July (first two weeks), but not available last two weeks of July, but would be available beginning of August again?)
I would prefer a two-week intensive course, in small groups or on a one-to-one basis. I am very active in a number of different sports, particularly badmington.
(They have asked about hobbies and interests to find out more about you as a person - they are not offering sports as part of the curriculum, so you don't need to say "..play badmington on some evenings if this is possible."
I can be contacted at the above address or on 420 723 604 067 at any time.
I look forward to hearing from you. (I would omit "soon", or even the more formal "at your earliest convenience".)
Yours faithfully (omit comma)
You will notice that I changed the wording to "would be available" and "would prefer". This is more polite, and not presuming you will be going to the school, as really, this is all conditional on being accepted into a course. In effect, what you are saying is:
If I was accepted into a course, I would be available
If I was accepted into a course, I would prefer a two-week
Student or Learner