Well, when I watched the movie, I was stumbled by this too. In reality, they used the movie footage. But, since the Klingons were in the council room, it's logical to assume that the klingon bird of prey took the footage.it was recorded by the Klingon ship
Yes, why wouldn't Klingon ships have exterior-view flight recorders? The question should be, didn't those records go with the crew to Vulcan, after leaving the Genesis World?
With Genesis being a restricted area the Federation could of had sensor platforms in the area that could record what was happening.