This kind of thing used to be done in telecine but these days, at least on a show like this, it'll be Flame or similar. Bear in mind that some of what you perceive comes from the stuttering they've introduced to the playback but other than that it's fair to say, that there's nothing here that you couldn't do in PS or Gimp. Some footage has been treated differently to others but it all looks like fairly basic grading. It doesn't look to me like much more than some levels adjustment and colour grading that's introduced some curve crossover. You can do this with the curve tool but I've also had good results with the colour balance tool where you can mess with correction and gradings in highlight, mid and shadow areas. Start off with cool (blue) shadows and warm up (yellow going red) the highlights and mid tones then adjust the result with the levels tool. Don't compensate luminance in the balance tool.
You'll need to have shot in RAW with a camera that supports at least 10 bits of colour. You'll be pushing it quite a lot and 89 bits will band.