(My answer assumes that you are in the US, or another country with a similar tenure system.)
Summary: Ask your local experts.
Every reputable university should have written documentation of the policies, procedures, and standards for tenure-track faculty promotions. Here are the procedures for my university. Read yours. Today. Now. Really, you should have read it before you accepted your current position. These are the policies that explain what "tenure" and "full professor" means at your institution.
A standard component of most tenure/promotion processes is that the candidate submits statements summarizing their most important accomplishments, ongoing efforts, and future plans in research, teaching, and service. The purpose of these statements is to answer the question "Why do you deserve this promotion?" These statements are almost certainly what your dean meant by "motivating your candidature". The specific format and contents of these statements depend on your university's promotion expectations/standards/practices/priorities, which is why you must read those policies before you even think about applying for (or even working toward) a promotion. In particular, research-intensive schools have different (not "higher" or "lower") expectations from more teaching-focused schools.
Another standard component is an evaluation of your tenure/promotion case by members of your promotions and tenure ("P&T") committee, which is almost certainly composed of senior faculty in your department, or larger unit if your department is small. The purpose of their evaluation is to answer the question "Why does your colleague deserve this promotion?" You should already know what they're going to write, because you've already discussed your case in detail with at least one senior faculty member who serves (or has served) on the P&T committee. (If you don't already have a senior faculty mentor, march into your department chair's office today and demand one.) Your faculty mentor knows the unwritten priorities/standards/politics that the committee will apply, and so they can effectively help you craft your pitch.
Another standard component is a letter from your department chair to the dean (or whatever titles your boss and their boss have), summarizing the strengths and weaknesses of your case and the deliberations of the P&T committee. The purpose of this summary is to answer the question "Why does your faculty member deserve this promotion?" Aside from snippets from your recommendation letters (which haven't been written yet), you should already know what this letter will say, because you've already discussed your case in detail with your department chair. In fact, you've already had at least one discussion every year with your department chair about your evolving career and your progress toward a strong tenure/promotion case (if you have not had such a discussion with your chair, walk into their office today and demand one). Your department chair knows the unwritten priorities/standards/politics that your upper administration will apply, and so they can effectively help you craft your pitch.
In particular, your university's promotion policies, your faculty mentor(s), and your department chair can tell you the expectations for a direct promotion from assistant professor to full professor. Such a promotion would be completely unheard of at my university---not impossible in principle, but never actually done in practice---but may be more acceptable at yours.
If your department chair and other senior faculty are not willing to have regular frank discussions about your potential for promotion, polish your CV and look for another job. At a rank commensurate with your accomplishments, of course.