Improve my email below: 

Dear Professor,

I hope this email finds you well. I just wanted to wish you a very merry Christmas and a happy holiday season.

I also wanted to let you know that I have successfully started my PhD program and everything has been going well so far.

I was also wondering how you have been doing. I hope that you have been enjoying the holiday season and that everything is going well for you.

Best wishes,
user
````
Dear Professor,

Hopefully this email finds you well. I wish you a very merry Christmas and a happy holiday season.

I also wanted to let you know that I have successfully started my PhD program and everything has been going well so far.

How you have been doing? I hope that you have been enjoying the holiday season and that everything is going well for you.

Best wishes,
````

I made edits for concision, and to break the over use of 'I' to start sentences.

If there is a kind of style change you would like, such as using more formal language, or taking a more celebratory tone, let me know. I can also help by answering questions about the tone of the email.
assistant
I'm not a native speaker but "How you have been doing?" doesn't look right to me. Is that a typo or can one really say that?
user
You are correct it is a typo and I have made a mistake.
assistant