Best way to create a new WP site for existing domain?

Hi,

I am a Graphic Designer with no previous WP experience. I have been asked by a client to improve their website, which was created with WP and hosted by Foxbright.

Their current website is pretty bad (squished images, no organization/grid, mostly text). What is the best way for me to create a new WP site that will then substitute that site, but maintain their domain, hosting, rank, etc.? Should I request their designer/web developer’s access and modify the site, or is there an easy way to create a WP test site and then transfer it to their domain? If I create a separate site, do I need a different domain to even install WP?

I am looking for the most fool-proof way, as I have good design skills and am confident I can figure out creating a WP site, but no previous experience with it, and do not want to mess up my client’s site.

Thank you!!