Quote from StratoFoxgloveAwesome idea, except I have no idea how this could be made possible...
As already mentioned it'd probably take up a lot of storage on Blizzard's end, unless the video is stored client-side and is inaccessible from the UI.
Another issue is: how would the client know when to start recording?
This problem is solved with crash-cams and things of that ilk. Simply, it is constantly recording but it only writes out to disk permanently when an event occurs. However, this is obviously not a practical solution in Diablo, as there are hardware limitations. Not everybody has a machine that can afford to constantly record video. Fortunately, there's a (relatively) simple way around this, and you can still use the same basic concept as before. Instead of recording video, you can log events, and simply "replay" the death using the game's engine to parse the events. This is how Quake (and other games) do "demos" which you can run from within the game. It simply records every event that happens and then uses the engine to simulate what happened. Now instead of megabytes of video you have kilobytes of text.