How to upload video from iPhone to Dropbox?
This worked for me just now (after downloading Dropbox):


Open Photos
Selected a video
Clicked the Share icon (square with arrow going up, left side)
Clicked on "More..." (scroll to the right, if necessary)
Turned on Dropbox and clicked Done (top right)
Selected Dropbox, filled in the info, and pressed "Post"


That should do it.