Is this the correct way to do it ?
This is not the correct way to do this.
What you are doing, first, doesn’t make sense conceptually. When you load that page (assuming it loaded correctly) you would be loading the entire WordPress page, not just the comments. That is, you will load <html> all the way through </html>. There is no point to that. All you want are the comments.
Second, WordPress provides a mechanism to centralize AJAX requests. It is called the AJAX API and once you get the hang of it you will never try anything else. Use that.
There are numerous questions on this site about using the AJAX API, including this one by me.
Do some research. Try to do this right. Then edit the question with particular problems that you are having.
Edit:
I took a look at your edit and…
You need die() at the end of your Ajax callback, per the Codex.
If my_ajax_script.ajaxurl is not set correctly nothing will work. Look at the source of the page and confirm that.