One way is to use a file management app like ESFileExplorer, navigate to the movie file (wherever you pasted it) and select it. ESFileExplorer will ask you which app to open the movie with. Select which app and the movie starts.
Yes I know, its kluge and cluncky like most things associated with the pitiful Fire interface and OS. The Fire does fairly well with Amazon purchased content, but it is can be a PITA with non-Amazon content.