If you post music videos online for the public, you need to be aware that the music is almost certainly under copyright protection and there are legal risks. That being said, I am showing videos on this site that sometimes include copyrighted music. I'm taking the position that these represent fair use, because (a) these are my personal videos, created for my family, and i am only showing them here to demonstrate how your personal videos may work with music; (2) these are not for commercial use (i'm not selling them, they are not being used to sell anything; and (3) i'm an educator and these are being used as part of my educational curriculum.
I seriously considered using more generic or public domain music, but i felt that while these would somewhat adequately demonstrate some aspects of how to use music with your videos, i was afraid they would be less inspiring and informative simply because they weren't real personal sketches, and that content constructed for teaching as opposed to my "real" content would be lacking in its effectiveness to inspire and inform. I believe that part of my value as an educator is through exactly this kind of candor from me.
That being said, if any copyright holders have issues with the presentations here, contact me directly and we can remove the particular sketch from the internet immediately.
Post Script: This PDF is to a useful checklist about Fair Use of copyrighted materials, from the Copyright Management Center - Indiana University/Purdue.