It took me way too long to figure this out, but there is no documentation at all on Sony’s store website, and google kept referring me to Windows 10 setups. We’re running a Windows 7 gaming setup, so things are quite a bit different. I’m putting there here so others might have something so search for.
To pair a DualShock 4 Controller with the DualShock 4 USB Wireless Adapter first, plug in the adapter to the PC. Windows 7 will automatically install the adapter and show it as a gaming controller. Next, push the Wireless adapter in a bit further and hold it until the blue light on the front starts blinking rapidly. On your DualShock controller, hold the “Share” and “PS4” buttons down at the same time until the light on it starts blinking rapidly. The lights on the controller and the USB adapter will then go solid. Your DualShock controller is now paired with Windows.