I had the same problem with the black screen after opening FCP 6.0.1. Not sure if these steps will be the same for FCP 5.1, but hopefully this may help. After starting FCP I moused to the top of the black screen. After a few seconds (maybe 5 seconds) the top menu bar dropped down. Clicked on Final Cut Pro and moused down to Audio/Video Settings in the drop down box. Then clicked on the Summery tab. From within the summery tab I clicked on the Video Playback setting. I had accidentally switched this setting to Digital Cinema Desktop Preview – Main. I reset it to None and clicked OK. No problems since.
Hope this helps!
Austin