Your PS4 is using your PC as a proxy, so it needs to be able to connect to it. As long as both are on the same network, it should work. For example, the PS4 is connecting via Wi-Fi to the same router that your PC is plugged into with an Ethernet cable. Or, both are connected via Wi-Fi to the same router.
The IP address you want to enter on the PS4 is your PC's IP address. By the way, the "checkIP" or "IPcheck" is just a batch file that opens a command prompt and runs the IPCONFIG command. You don't need it, just do that yourself.
If they are not both on the same network, it won't work. Also, the proxy program (Squid) can be blocked by a firewall. If you get a Windows Firewall message (as I did) about whether to allow Squid, you MUST answer Yes. If you didn't, you have to go to "allow an app through Windows Firewall" or "allow a program through Windows Firewall" (depending on your Windows version) and add the exception manually. If you have a 3rd-party firewall on your computer (Norton, McAfee, etc.) you have to allow it in there.