How do I create footage that has nearly the same quality as https://www.youtube.com/watch?v=iNJdPyoqt8U but is taken by an action cam?
The Simple answer is going to be "No" That may sound harsh but you cannot as there are so many factors that were involved with this video unless you want to buy similar kit to the guys who made the video. The main thing being the camera and lenses they used
Tech spec:
Red Epic (M642) w/ Ti Canon Mount
Canon 24-70mm F2.8 
Canon 16-35mm f/2.8
Canon 100mm f/2.8
Canon 70-200 f/2.8
Adobe Premiere CC and DaVinci Resolve
Freefly System MOVI M10

Your action camera, while it can record 4k video cannot compete with RED RAW files nor can the lens compete with the dedicated lenses listed above.

Now to answer the question: to achieve this type of video quality would be to invest in a better camera that can use different lenses and has good quality video capabilities, now having a RED EPIC is great but also having a HUGE budget is great but there are small decent cameras that can record 4k like the BMCC or Panasonic GH4 or Canon C500 etc but besides having the best kit in the world you would need to know how to use it all and probably have a team to help.

With the post-production any decent NLE can achieve this but it is knowing how to edit that is the skill.

Hope that helps and doesnt sound to bitchy