In the latest firmware update (3.50), Sony have added the ability for you to access the Playstation 4‘s remote play feature through a PC. This means you can now access the same remote play that the Vita and Playstation TV can, with your PC. Here is what you need to do to setup PS4 remote play on PC.
Download PS4 Remote Play PC Software
You can download and install the software from the official Sony website here. https://remoteplay.dl.playstation.net/remoteplay/lang/en/index.html
Once the software is installed, run it and it will ask you to sign into PSN. Do this and it will attempt to find your PS4 over the internet. It is likely going to fail for whatever reason, but if it works, then you are done here. If it fails you will have to click register manually.
Manual Registration Code
To manually register you need to get the manual registration code. Go to your PS$ and head to the settings tab. Go down to “Remote Play Connection Settings”. Click on the “Add Device” menu item. This will display an 8 digit code on the screen and give you a time to use it. Head back over to the PC and enter the code onto the screen. This will register the connection with the PS4.
Sometimes this connection will generate an error. This will most likely be patched in the future, but for now, if this happens restart your PS4. Once it has restarted you should be able to reconnect to the PS4 and it should work without a problem.