The PS4 can be tricky, especially since its so precisely engineered. Replacing the Thermal Paste may help, but the fan going hyper can also be a symptom of a bigger problem. I only recommend doing this if you have an ESD safe work surface, and are comfortable taking your PS4 apart.
You can find the guide to taking the PS4 apart here. Follow it until you remove the heat sink, then use this guide to remove the old thermal paste, and apply the new thermal paste.
Here is what you will need:
- Thermal Paste (Artic Alumina should be fine, but you can choose something higher-end if you want)
- 2 Step ArticClean solution for removing the old paste, and prepping the service for the new.
- iFixit 54-bit Screwdriver Kit (I like to recommend them over the 3rd-Party brands since they tend to last longer anyway, and they have a Lifetime Warranty on their tools.)
- Precision Tweezers for removing warranty stickers.
Bare in mind that if you do this, you will be voiding the warranty, if it hasn't expired already. Now that the disclaimer is out of the way, have fun!