There are quite a few ways to build build a new WordPress website while the old site is live, but this is the easiest non-technical way to do it.
Original I wrote this post in 2014 and it’s still my preferred method today.
Step 1: Install a new WordPress website
Install a new WordPress website. For this method don’t install in a sub-directory of the existing website. Just create a completely new website!
Step 2: Re-point your domain name
In this step you need to repoint your domain name to the correct server IP address. Once you’ve done this you need to wait for the domain name to work i.e the new website will start showing for the domain name. Once this has happened then move to step 3.