Whether this is possible at all will depend a lot on your stage of training and how much time you can devote to being involved in a project. Assuming you are already "in the system" in an academic medical center (med student or resident, etc.), you could probably find a role in a trial being done at your institution. Contact the research coordinator in the division of Your Favorite Subspecialty and see if they need someone to push some pencils in a local trial. You might be able to review charts, complete paperwork or do something like that and thereby get your name on the eventual publication. It could be pretty educational to see the inside of a clinical trial and could help you figure out if you want to do this type of research in the future--but having something like this on your CV will not exactly make your career.
The other type of trial is a larger multi-institutional study; these are the high-profile papers you see in NEJM and JAMA every week. Authorship on multi-institutional trials is highly political and if you are asking this question, it is probably out of the question. Each institution enrolling patients gets to put in a couple of authors and their order will depend on how many patients they contributed. Almost all of these people will be attendings. Then the sponsor will throw in their own study pathologists, statisticians and whoever else was involved in the central parts of the study. Medical students and office staff who helped fill out paperwork might get mentioned in the acknowledgements if they are lucky.