FH comments- You seem capable of generating pretty decent pace. I bet you could get another 25% more pace out of your shots with less effort based on what I see. The key is, you guessed it, to relax and find the fluidity in this stroke, much like you see when Federer hits a FH.
So when I watch you hit your FH, it still looks stiff and almost like you are "arming" the ball. You want to be uncoiling as you are contacting the ball. If you watch closely, you will often be open already while striking the ball (i.e. chest facing net), so you are relying on mostly arm strength here (i.e. almost slapping it) instead of using the stored energy from your legs and core rotation. Sometimes you are even falling backwards a tad, or to the side on your follow through instead of moving towards the target. This tells me you have spent some of the potential energy you stored up in your legs and core prior to contact. This is one source of power that is lost and not being transferring to the ball. Probably on the order of 15-20% is lost.
Another inefficiency I see is in your swing. What seems to be lacking is a certain fluidity. Your swing seems like it is more horizontal than vertical at contact, almost as if you are hitting it with your wrist and forearm straight out, like a baseball bat. Instead, you want to have your racquet swing more downward and then extend towards your target after your takeback. Try and use your left hand to help you set up (i.e. pretend like you are catching the ball with your left hand).
With the face of the stringbed facing the back curtain, find the swingpath that approximates a pendulum and try and keep your frame in this plane as you swing towards your contact point. Try and lead with your wrist as you start to drop your frame towards the bottom of this "slot" and continue to lead with the wrist coming out from the bottom. As your frame begins to rise up you should be uncoiling from your legs and core. With you arm about half way between the bottom of the slot and the contact point the racquet head should now be accelerated using your wrist (i.e. think of a the racquet as being the tip of the whip in this kenetic chain). If you time it correctly your forearm should be extended towards your target on contact and your wrist and racquet's stringbed lined up in the same plane perfectly perpendicular to the vector that represents the balls desired flight path.
As you make contact continue extend along this vector before breaking your swing back across your chest on your follow through. Your chest should be facing the target during this extension period and your left arm should on its way to being tucked in on your left side as you allow your racquet rotation to finish across your chest. Your right leg may come around as you weight transfers through the shot and you move towards your recovery step.
If you can try to find this motion without hitting a ball it might help to practice it that way first. Then when you go to hit a live ball, start out by just trying to groove this fluidity without worrying about how much pace you are generating. You should be completely relaxed while striking the ball, much like the pro's look like when they are warming up. Once you've grooved the stroke, you can then start to work on generating pace through the proper core rotation and acceleration of the wrist and forearm through contact. The key to power here is to remain relaxed through the entire stroke. I guarantee you that if you find this, you will be able to hit a heavier ball with significantly less effort than you are expending now. You will then have an extra 20-30% additional power in reserve to really crack that FH.
I'm pretty sure some of the teaching pros on here can give you a better description and more helpful pointers than I have here. Hopefully they will chime in and add their comments too.
I'll look at the BH a little later on....
EDIT --> The FH's at 2:24 and the one after that look better. The BH looks really good. I'm not a 2H'er so I can't offer much useful advice there but it looks solid, good acceleration and extension. We need to see some of this on the FH side!!!!
