My grip is quite wide here with my index over the outer smooth ring. I was using a narrower grip before but I found this caused a few problems for me. Specifically, I think I ended up having the bar too high on my back, causing the bar to roll up my back sometimes and also my centre of mass to be further forward when i squatted down. Secondly, I ended up having my palms against the bar and my fingers over the top to allow me to have them closer, which I think Mark said was an incorrect grip and may have caused my elbows to hurt also.
My hips seem to move up without my upper body at the beginning of the ascent but I don't know if this is actually a problem or if I could easily fix it.
Only other thing I notice myself is the squats look a tad high on the video though I'm not sure. I thought I was going deep enough while doing the set.
Your squats do look high. Some of them by two or more inches. The solution here is to drive your knees out more and commit to going deeper. Keep your chin down and keep driving up with your hips. You have a hint of lifting your chest. Grip looks fine, although you could probably drop your elbows a bit. On the whole, decent squats, just go deeper.