Instructions
Open the WoW launcher by double-clicking the game icon on your desktop. Click "Public Test" at the top of the launcher. A drop-down menu will appear. Click "Download Public Test." Your browser will launch the Battle.net login page.
Enter your Battle.net account details and sign in. Once the next page loads, click the blue "Download the Client" link. Click "PC/MAC." The "Download Games" screen will appear.
Select your operating system next to "Public Test Realm" to download the client. Launch the "PTR.exe" file once it finishes downloading to install the client. While it's installing, go to Battle.net and access your WoW account. Under "Additional Services," click "Public Test Realm" to copy a character from the live servers to the public test realm.
Open the WoW launcher again after the public test realm client has been installed and you have copied over at least one character. Click "Public Test" and click "Play Public Test."