Video quality means a great deal to me. It made me switch from Canon to a Sony A33 way back when, because it provided continuous focusing while the Canon did not, making the Canon 100% pointless to me for video. Since then, I've discovered that it's more like Minolta 2.0, with top notch IQ for photos, and extras for video. The video contribution is Sony I feel, because they've been really good at it for pro equipment.
Then after than I switched to the A57, because the stabilization worked much better on it, plus 60 fps video, making it smooth. 60 fps is smooth. Anything below, is not. Don't belief for a split second, anyone claiming otherwise.
But then I got annoyed with the graininess of the A57 video is even moderate low light. I tried an A3000 as a second camera for the wife mainly, and there already, it showed that E-mount was so much better than A-mount for video. Smooth silent quick auto focusing, and better in low light. Then got excited over the A6000, and to me, it had "major milestone" written all over it. And it turned out it is. For the price, look what you get, it's pretty much *the* best deal there is, period. Image quality is very good, low light is good, resolution is good, tons of auto focusing features, it's *fast*, and video is great on it!
I think that for video it does full sensor readout - which is a biggie, because it means very good quality in low light. Sharpness wise, it doesn't cheat. What I mean by that, is that it doesn't apply a sharpening algorithm to make it look better than what it really is. Samsung does that for instance. But who are they fooling? Not me! Sharpened video is more difficult to compress and causes much more artifacts! If you want sharpest playback, it's WAY better to sharpen during playback. For that, I use mplayer. That's on Linux, but there mplayer based players for the other platforms as well. It can sharpen in real time, at playback. The end result can be just as crisp as Samsung video, except, then, without the artifacts.
You need a good lens for video. My favorite is the 18-105 f/4 G lens. It is f/4 constant across the range. It's nice and sharp. Much higher quality than an 18-200 lens. It has power zoom and also manual zoom. The stabilization which is always lens based with E-mount, is very good!
I think that the *only* camera that does video better would have to be the GH4, but the cost is absurd on that one.
Get the A6000, I highly doubt you'd be disappointed! If size is ok, skip the kit lens, and fast forward straight for the 18-105 lens! It's definitely a bit of cash, but it's nice, and it's good!