Daniel, the beginning of what you do in the video is much like the old time side press. In that move, the stance is wide enough to allow the lifter to bend over while keep both knees locked, and although it's hard to tell with long pants, that's what you do for quite a while in your press.
If you look on my site, you'll find an old video of Bryce Lane doing a barbell side press with 135 lbs.
http://kbnj.com/pressed.wmv
or
http://kbnj.com/pressed.mp4
What you need is to buy the book, Taming the Bent Press, by our own Iron Tamer David Whitley, and follow its instructions. You will want a narrow stance and to bend the leg on the non-bearing side.
Nothing wrong with a side press, of course, but just as one can move more weight with a side press than with a strict military press, so can one move more weight with a real bent press than with a side press.
Hope that helps.
-S-