I have the exact same graphics card, and I have had good luck with it so far. It works out of the box with the built in radeon driver. That driver supports the card surprisingly well. 2D & 3D acceleration works to some extent (including smooth performance in Compiz and Gnome Shell), but you'll have less luck doing 3D games or the like. On the bright side, Minecraft doesn't work very well with the free driver :b
So, to gain those other things (including making Minecraft work very, very well) you can install the properietary fglrx driver. This is available right away from the Additional Drivers tool. Just open System Settings and Additional Drivers will be listed in the Hardware section.
The fglrx driver does good 3D acceleration and all the bits in between. It definitely meets the two requirements you listed, as far as I have noticed (though, to be honest, I don't think I have played a full-length 1080p video thus far; just little things).
Unfortunately, the fglrx driver isn't quite perfect. It has a bug that breaks Gnome Shell, so that's why I'm currently using the built in driver. If you don't care about Gnome Shell, that bug won't be a problem.
It's pretty unusual that video playback in Ubuntu will use the video processing hardware built in to the GPU, so any claims of "1080p video playback!" are probably not so clear-cut. The video hardware in the Radeon HD 6800 series can be accessed through VA API, which is still somewhat new. We're still waiting for VA API support to land in GStreamer (the standard media framework in Ubuntu, used by the default Movie Player), but you might have luck if you install the VLC Media Player, which reportedly does VA API right now!