How to Use PS4 Remote Play

PlayStation Vita

Remote Play allows PS4 users to play console games through their PS Vita. After updating software on both platforms to the most recent firmware, you will find a new prompt on the PS Vita that states "PS4 Link". Clicking on this will search for an active PlayStation 4 in proximity range.  You must allow your PS4 and Vita to link up first, however. Go into your PlayStation 4's Add Device screen to find a temporary code that you can input on your PS Vita device. This will grant access across the two platforms, allowing you to instantly peruse the Remote feature. This is a one-time ordeal, which will never have to be done again between the two specific pieces of hardware.
Syncing the two devices requires a sturdy WiFi connection, but is seamless. Once synced, users can control PS4 games with their Vita, playing using the small screen. Controller options will reset upon the switch back and forth, so diving into the Options screen of the game you're playing may be required. 
Remote Play has a limited but undefined range. The contingency with this feature is that both the PlayStation 4 and the PS Vita be located within the confinements of the same WiFi network at the time of gameplay. Taking the Vita outside of the range of your network may result in connectivity problems. 3G will not support Remote Play.

Companion App

The Companion App is a secondary application that you can download on your Apple or Android smartphone device to use as exactly that: a companion to your PlayStation 4.

Go into your smart device's store and search for the PlayStation App from Sony Computer Entertainment. The app is free and thus just needs to be installed. Accept the user agreement and you will quickly find the Connect to PS4 button on the lower left-hand side of the app. Click on this to search for a nearby PS4. On the PlayStation 4, dive into the Settings > PlayStation App Connection Settings > Add Device, which will follow similarities to the Vita connection, where you must enter the temporary code to connect the devices.

From here, you will be able to purchase games through the PlayStation Store on your mobile device, or even use your smartphone as a secondary screen for in-game uses or as a secondary keyboard for typing messages, sending friend requests, and more. Enjoy your interconnectivity between media formats.

PS Vita As A Second Controller

If you only have one controller for your Playstation 4, you can use the PS Vita as a second controller. Start the PS4 link app on the PS Vita and select Remote Play which will make you login automatically into your PSN account, now press the home button on the Dualshock 4 controller and login as guest, if you login to your PSN account on the Dualshock 4, you will automatically lose connection.