Submitting a paper to a new journal with responses for the rejection to the first journal

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.

Don't do this. What'd you actually highlight would be:

• Your paper has been rejected by another journal before (not a good sign; think of it as "if we publish it now, we're actually publishing another journal's rejects, and we're supposed to be a good journal!!")
• You're not acting professionally. It seems like you've simply taken your submission to another journal and submitted it to ours, without having edited it to fit our requirements. If e.g. your cover letter also says "to the editor of journal A" and we're not journal A, your submission would be in trouble.
• We can't use the original journal's reviewer comments & your responses. We don't know who the reviewers are. We can't tell if the reports are legitimate. We can't see if confidential comments were submitted.

Let your paper stand on itself, and don't send additional package with it.

Keep this very clearly in mind: The referee reports from your initial submission should be considered, by default, as confidential communications between you and the initial journal. You cannot send them to the new journal without explicit written authorization from the editor of the initial journal.

(This isn't quite a hard-and-fast, always-the-case standing imperative, but you should think very carefully about how and why you break the confidentiality of referee reports, and you should read carefully any statements by the initial journal regarding confidentiality. If the journal tells you that reports to are confidential, you break that, and they find out, they won't be pleased and you'll have a rather harder time the next time you submit to them, if nothing else. And there's a nontrivial chance that the new journal will be less than impressed by the breaking of confidentiality.)

Thus, you should weigh very carefully just how much useful information you can transmit in a response when you're not allowed to quote the comment that you're responding to. In principle it might be possible to broadly describe the types of concerns that were raised and then go on to talk about how you've edited the manuscript to fix those, but that letter just isn't going to be very effective.

All of which, of course, is over-ridden by the fact that this is a terrible idea to begin with, as already explained by Buffy and Allure. If the initial submission's reports were useful in pointing out flaws in the manuscript, then by all means fix them in the manuscript.

(In particular, if the reviewer pool for your topic is small and the new journal independently selects the same referees -- by no means an uncommon occurrence -- then those referees will be pleased to see that you've addressed their previous concerns. It happens with some frequency that referees will reject a paper on journal A and then get contacted by journal B to referee an unchanged manuscript; nothing spells death on a journal submission quite like that.)

Remember: the reviewers are part of the audience for your paper; in fact, they're the only members of your paper's audience that have both an explicit incentive to read your paper carefully and care about its contents and correctness, together with the opportunity to communicate all of their concerns to you. If there are parts of the paper where they didn't understand, or where they raised arguments which you would like to engage with, then address that in the text of the manuscript itself, where it will be available to the future readers who, like your reviewers, are likely to have similar concerns. Shifting those responses to a cover letter where they will be invisible to your audience does your readers a disservice.