The first things I'd do is figure out where the users, their previous scores, questions and answers will be stored (probably a database). Next, I'd create a HTML template for the site and figure out how the quiz will work. For example, will it show 1 question at a time or a bunch? Will the user be able to see their correct answer immediately after each question or only at the end? Will the user have to complete 1 quiz in order to see the next, can they scroll through quizzes or is there just that 1 quiz? Will the users be able to freely type answers (i.e. sanitizing their inputs) or will the choices be given to them?
In terms of the user profile, you're going to have to ask yourself what is it going to show. Will it be just the user and their score or other things, such as uploading a photo/avatar, a blurb about themselves and other stuff that online profiles on sites often have?
Perhaps sketching out on paper how you want the site to look, even if it's just a rough draft, I find gives me some idea of what I'm trying to create and prevent myself from going on a tangent (even if it's a fun one).
I cant suggest to you what to do for your quiz as you have to start it yourself.