Creating a WordPress website using an existing domain

I was wondering if anyone could explain something for me as simply as possible.
I have no computer background but I have my own www.remarpro.com website and found it relatively easy to set up and design. My parents own a clothing store and their current website is not very good, to put it simply, so i offered to create a whole new site for them using wordpress. I don’t need to copy over any existing information or anything from the old site to the new as it needs a total overhaul i believe it would be easier to just start from the beginning again.
So what is the quickest and easiest way to create the new wordpress site using the same domain name without the website going down or customers being able to see the new site as i’m working on it? It would be great if customers could still view the old site whilst i work so it doesn’t impact the business too much. So far i haven’t installed wordpress or done anything so starting from square one here
I have searched a lot of the forums and read a great deal of information but people seem to be more focused on moving content over.
Any information is HUGELY appreciated as i only have a limited amount of time to attempt to create this website!
Thanks in advance
Katie