A grammar question.

Hello, I want to ask a teacher if I can join a certain level class. Here is the sentence I thought of using, please tell me if it is correct:
Green先生に書いているのは私はJ３０１クラスに参加させていただきたいです。
Thank you!

Nick, Are you going to tell it to your teacher in person or send a letter/note to him/her?

Green先生に書いているのは私はJ３０１クラスに参加させていただきたいです
I don't think the first section of the sentence (Green先生に書いているのは）is not necessary.
「Green先生、J301クラスに参加させて下さい」or 「Green先生、J301クラスに参加してもよろしいですか？」will do.

Guessing from your sentence, what you want to say is something like:
"I am writing an e-mail (or a letter?) to you, Mr/Ms. Green, because I would like to be allowed to join J301 class"
...right?

To *write* that, you would want to say:
Green先生にこうしてメールを(or手紙を）書いているのは、J301クラスに参加させて頂きたいと思っているからです。
You might want to add something like:
いかがでしょうか。(What would you say?)

Ah, yes, I am sending a letter. Thank you, those suggestions look much better than what I had written. I appreciate the help.