Submitting a paper to a new journal with responses for the rejection to the first journal I submitted my paper in IEEE Transcations of XXX and got a rejection. But the comments were really helpful and has helped me a lot in improving the paper. But, unfortunately, I cannot resubmit that paper to this same journal again and now I am looking for another journal. 

I am writing a proper response again every comment and thinking to submit the response letter as part of the submission to the new journal. The basic idea is to let the reviewers know that this paper has been reviewed (by reviewers of another journal) and improvements has been made. Will it somehow improve the chances of acceptance. (as I am in the final stages of my PhD now)

What does the community say?
I wouldn't do that, actually. Use the comments of the reviewers of the first journal to improve the paper as usual. Simply submit the revised paper to the other journal. 

It is actually better that they look at it with a fresh viewpoint, rather than having their view possibly clouded by the opinions of others. Let your paper stand on its own. 

You will get additional comments, of course. Hopefully they will also help you improve it further.