iOS 9.3.1: Dropbox 6.2 doesn't appear in Activities in Photos
Go into Settings > Dropbox and verify photos is turned on. Now go into Photos, select a picture and tap the share icon. There are two rows of icons, you want the lower row (on mine it starts with "Copy.") Scroll the icons to the left side of the screen



If Dropbox is not there tap the "..." "More" button and turn on Dropbox.

That's how it worked on my iPhone 6+ with IOS 9.3.1.