I'll talk about broken octaves generally rather than respond specifically about the video link, since it may not stay up.
Broken octaves are a standard part of piano technique from Mozart onward.
Forearm rotation is the major source of power in playing them. I suggest the following simple exercises to get used to it, if it's new to you.
Firstly, try playing a slow, mezzo forte tremolo on C in the left hand, that is, C with the little finger, then C with the thumb, then C with the little finger, and so on. Exaggerate the rotational movement: rotate the forearm far to the left to put 5 on the lower note, then rotate far to the right to put 1 on the upper note, and so on. Gradually increase the rotational movement up through forte to fortissimo. If done well, you should manage fortissimo with absolutely minimal effort.
Secondly, you can move on to broken octaves proper, playing up and down C D E F G F E D C. Here a little more practice is needed due to the changing position of the hand. Continue to exaggerate the rotational element until you're quite sure you've "got it" and are achieving an effortless forte.
Later, when you need speed, you can look at minimizing the rotation - as speed is all about minimizing unnecessary movement - but it will always be present.
Finger action is involved in broken octaves too, but for now look upon it as fine adjustment to make sure that the right key is played, not as the source of power.
With practice, you'll find this forearm rotation can be done very fast and effortlessly. It can be used for trills, tremolos, broken octaves, and for power in non-repeating patterns too.
Specific advice about the pains you have suffered shouldn't be given without observation. Generally speaking though, don't open the hand wider than necessary, and don't tilt it toward the thumb or little finger. That is sometimes necessary, but not for broken octaves unless they reach extreme positions on the keyboard.