I'm designing a website and need to know how to give my client access to viewing the site before going live.

You're going to have to have some place to upload the site to the web.
Do you have personal webspace ?? You could make a temporary subdirectory for your client's files, give them a link to that subfolder and they should be good. Once they see it, you take it down off your web space.

Or if you are building the site on their domain put in in a password protected directory so the public can't access it.

I usually set up a sub directory on my site where I'll build the clients site. Depending on the site I may or may not password protect the sub directory. That's probably easiest if you have your own site.