In most cases shadow detail is what is lacking and the picture looks very dark and even unpleasant sometimes. Video codec, aliasing, lack of resolution etc. makes things even worse. This occurs mostly when shooting at available light outside and sun is pointing somewhat towards the camera (or some other light source doing the same). Even when it is not directly pointing to the camera, the areas on front of it will be very easily dark. Sun is extremely powerful light source and competing with it with exposure can be tricky.
People usually shoot with flat picture profiles, which a little bit helps on this case too. However, flat picture styles will reduce the available dynamic range because it compresses it. You will get more shadow details because the blacks are moved to right in the histogram, but the image quality reduces.
I found an old article which recommended exposing towards the right side (towards highlights) of the histogram. I could not believe it because I was thinking that I will lose dynamic range if I do that. I made an experiment: I shot an evening scene which was exposed as dark as it was naturally to the eye. Looks like the right way to do it? Yeah right. I took another shot which I overexposed it so that the histogram was leaning to the right (highlight side) so that everything was too light still without clipping the highlights.
As expected from my experiment: the scene that I made to look like it was in reality looked like it was in reality. Fiddling with Aperture - there was some noise and I could not get much more shadow detail out of it than there was. It was like it was exposed to look like it was and that's it. No further alterations very much possible.
Ok now I then went to the next picture which was overexposed and a bit too bright. It looked like a failure to begin with. However, after I adjusted the exposure in Aperture, it started looking like it looked to the eye in the evening. And guess what - it had less noise - with same ISO setting! I could find lots of shadow detail in the picture.
I was recently also looking my old video scenes that I have shot for example in North Cape. In general they are rather muddy and dark. Some have rather unpleasant look. I have been trying to adjust them with Color and then with these couple of free plugins for FCPX but if I bring up the blacks, block artifacts and noise start to show up rather severely and the video quality drops to almost to a level of SD. In middle of these scenes there was one overexposed scene. And voila, when I exposed it in post processing towards blacks, there was lots of shadow detail, and even highlight detail in the scene and no visible noise.
So what I do now when I shoot next time (this is still an experiment, so don't ruin your footage based on this instruction, or if you do, do it with your own risk): - I use a bit less flat picture style than e.g. Technicolor because the picture style after all is not a holy grail and does not fix all the problems in the exposure there are because the file format is not 12 bit RAW. Neutral picture style with contrast reduction is fine. Another possibility is to use Technicolor Cinestyle profile with contrast setting in the middle (not dialed all the way down). - Then I expose a bit towards highlights but make sure the highlights do not clip (otherwise the image quality will be low because highlights are clipped).