first off, you have to have a hacked psp either X.XX M33, or X.XX OE. Second, when the psp starts up, you can only change the text where it says " Sony computer entertainment" and the sound. So no you can't put a part of the music video on startup. You can put part of the song though.
However, there is a gameboot( when you start the game it has the green and yellow circles and it says psp). You can also customize this. On this you can put part of the music video. You can ask for help to do this or look at some youtube videos.