Of course, I'm going to assume that you'll use your site analytics to identify a time period when you typically get fewer visitors to schedule the maintenance.
Given that, once you've decided on the exact timeframe for the outage (and allowing a little extra time for contingencies)...
Notify people ahead of time so they know to not try to transact business on the site during that time. Post a notice on the home page (and interior pages, too, if you think people will enter to those pages directly and not see the home page). If this is some kind of membership site, post it on the login page and on the first page that people will land on after they log in.
It doesn't have to be fancy, but it should be noticeable. Put it in bold, in a different colored text from the rest of the page, and put it "above the fold" (that is, toward the top of the screen, in an area that they will see no matter how big or small their monitor, no matter what sort of device they're using to access your site).
Post it sooner rather than later. The benefit is that it increases the chances that people will see the notification, and it makes it clear that the outage is scheduled maintenance, not an emergency problem with the site.
If these are clients, you should have an email address, right? You could also send them an email to let them know. It probably wouldn't hurt to send at least one, just in case there are some who don't visit the site between now and the shutdown or who do visit but manage to overlook the notification on the site.
During the actual maintenance, have a placeholder page displayed that lets visitors know the site is down for planned maintenance and that they should try to access it again later. If you want to get fancy, you can even include a "countdown timer" to when you expect the site to be back online. (Just be sure to allow some extra time in case of problems -- nobody is going to complain if the site comes back online sooner than you predicted, but you might get complaints if it's significantly later than you thought.) There are several Wordpress plugins as well as third-party scripts for non-Wordpress sites that offer this kind of functionality.
Does this answer your question, or was there something else you needed to know?