How do I connect through a HC-05 to PS4 Controller?
The HC-05 uses the Bluetooth Serial Port profile (SPP). The PS4 controller doesn't. 

Hence you need something that is capable of using the same profile that the PS4 controller uses (HID?). And to that end you need something a little more configurable than a simple HC-05. One option is, as most people use, a USB Host shield with a Bluetooth dongle that can then be configured to communicate using the right profile.