How to properly upgrade your website using under construction / offline

Im working on a site for someone who has an online store and although I've been doing a lot of web design jobs, I've never encountered or needed anyone who needed to have it done without altering their current site until the final look is established.

How would i create all the content offline or on a hidden site then transfer the site when I'm done?

Basically you would need a copy of the entire site (including databases) on your own computer, work on it offline, and then transfer it back online. Depending on the platform used, there could be backup softwares available that do this automatically. You need to install for example Apache server software to your computer though (for example WAMP or XAMP), so that you can properly view for example php files and the database content on your local machine.

On which platform is the online store built?

If its just about adjusting/altering the design with css and some images - you won't even need the database but only a copy of the website made with httrack.