Is the Insanity workout a good approach?
If you need to pay for your routine, no.
If it is, should I be following the diet it recommends (including the protein powder)?
Absolutely not. Since you mention that you are quite skinny, what I recommend is eating more than usual (caloric surplus). Protein shake should not be your main source of protein.
If you really want to make a change to your body, consider calculating your TDEE.
As far as protein intake goes, the usual 1g per pound of bodyweight. Fat intake probably around 0.5 ~ 0.6 g per pound of bodyweight. Fill up the rest with carbs, vitamins and minerals.
Do note that stated protein intake, fat intake is not your final answer. You may need to modify it to ensure that you are able to ensure the best muscle to fat gain ratio.
After having a long conversation with @Ryan, I would like to suggest that you opt for a free calisthenics workout program BEFORE buying Insanity. Once you feel happy doing calisthenics, then you can decide whether you want to buy Insanity or continue with your routine.
Here is some of the free ones for you to try:-
Nerd's Fitness Beginner Bodyweight Workout
Fortress Bodyweight Workout
Note that if you are unable to do pull-ups, consider doing incline pull ups and if you are unable to do pistol squat, try doing squats.