What is causing this chain clicking noise?
A couple of things I see:  


I think your derailleur is a bit out of adjustment and is "leaning" towards the lower gears.  It's grabbing on the ramps of the lower gear as it goes by.
Your B screw adjustment is a bit too far away from the cassette.  I would try to get it within a link or link and half.  It looks from the video that you are about 2-3 links away.  See section 4 of this link: http://www.sheldonbrown.com/derailer-adjustment.html


Lastly, is it possible that 9 speed chain (or less) is on that 10 speed cassette?  If the chain is too wide, it would have the same symptoms (grabbing the ramps)