This tutorial will teach you how to backup your PlayStation 4 game data to PlayStation's online backup service. Use of this method will require a current subscription to PlayStation Plus.
Note: If you do not have a subscription to PlayStation Plus, you do have the option of backing up your game data to a USB key.
How To Backup PS4 Game Data Online
As previously mentioned, the use of PlayStation's online service requires a current membership to PlayStation Plus. The price of these memberships range from $9.99 for 1 month, to $17.99 for 3 months, up to $49.99 for 1 year. Subscriptions to these memberships can only be activated from a primary console.
To activate a PS4 as your primary console, sign in to the PlayStation Network, go to Account Management > Activate as Your Primary PS4, and click on Activate.
How To Enable Automatic Uploads on the PS4
Once everything is set up properly, go to Settings > Power Save Settings > Set Functions Available in Rest Mode > Stay Connected to the Internet.
Tick the Enable Automatic Uploads checkbox and then select the games you'd like to add to the backup service.
N.B. PlayStation Plus subscribers are entitled to 10GB of free cloud storage on the PSN.