There are two methods for installing the update: directly onto an Internet-connected PlayStation 4 or with a USB flash drive. The latter can be utilized for those who don't have an Internet connection readily available.
Here's how to do it:
Step one: Insert a USB drive with at least 1GB of free space into your computer.
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Step two: Create a folder on the drive titled "PS4" -- no spaces.
Step three: Inside of the PS4 folder, create a folder named "UPDATE" in all caps.
Step four: Download the update from Sony's website and save it in the UPDATE folder on your USB drive. The latest update is version 1.50.
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Step five: If the power indicator on the PlayStation 4 is orange, hold the power button until the system beeps (approximately 7 seconds). Once the system is turned off, connect the USB drive.
Step six: Boot the PlayStation 4 in safe mode by holding the power button for 7 seconds.
Step seven: Plug the controller in, hit the PlayStation button, and select the "Update System Software" option. After a few minutes and a couple of restarts, your PlayStation 4 should be up to date.
Step eight: To ensure the update was properly installed, enter Settings, click on System Settings, and select System Information.
