I agree w/ everyone in that iTunes does not treat or allow music videos on iOS devices that were not purchased through iTunes and are not currently available in their catalog. That really stinks, especially that you can't view even ones you've bought through iTunes in the past.
However, I did create a workaround to allow for syncing music videos to my iPad/iPhone when I want.
1. In iTunes on your Mac/PC, select music videos (in your Music section's Songs listing) that do not sync w/ iCloud but that you would like to have on your iOS device.
2. Once you've selected the videos you wish to sync, either in a group select or individually, click Get Info either under the Menu > File or right-click.
3. In the popup window, select the Info tab then check the Genre box and change it to "Music" or a custom Genre you create like "Music Vids".
4. Next, select the Options tab and change the Media Kind from Music Video to Movie, then hit the OK button.
5. Now create a new Smart Playlist that requires All for a "Media Kind is Movie" rule along with a rule for "Genre is Music" (or the custom genre you created). Be sure that live updating is checked also. Name your new playlist "Music Vids" or something you like.
6. Finally, in your iOS device's sync window in iTunes, you will find an option under the Movies tab to allow you to include movies from certain playlists to sync to your iOS device. Just select your new Smart Playlist in the listings.
7. Now sync your iPhone, iPod Touch and/or iPad.
Viola...you now have your music videos showing under the Movie heading within the iOS' Video app!