You see an anime music video on YouTube, and think you'd like to make one of your own. If you aspire to produce your own video, then the first, and most important, step of the project is planning.
When it comes to the music for your video, if you or your friends are musicians, then you would probably want to seriously consider having a piece of music written specifically for your project. If you are wanting to use music that's copyrighted by another artist, then you are running the risk of violating copyright law. If you're planning to make an anime music video for your own enjoyment, then using someone else's music may not be a problem. However, if you are intending your anime music video to be seen by a wider audience (whether it is through YouTube, or by distributing copies to your friends), then you are distributing another artist's copyrighted material for free.
For video footage, it is recommended to come up with your own footage using a video recorder (for example, recording stills of fan art or footage of an anime property's merchandise). While many anime music videos use footage from the actual anime, that can technically be considered a form of copyright violation. However, if you're using just small clips, you might have a better chance of claiming that your video is a form of "fair use" and that you aren't intending to redistribute the entire anime. Like with the music, if you're simply making the video for your own enjoyment and not planning to share it with others, then this really shouldn't be an issue. However, if you distribute the video to friends or post it on YouTube, you could potentially be opening yourself up to "cease and desist" orders or legal challenges from copyright holders.
After you make your decisions as to what music you are using and what video footage you will be utilizing, you need to decide what footage you want to have shown at each point in the song. This is best accomplished by putting together a storyboard. When you're making these decisions, it's better to go with the feeling of the music and the lyrics, rather than trying to make literal matches between the footage and the lyrics (for example, using footage of someone crying every time the lyrics of a song mention crying or tears). Overall, the more you match the footage with the feeling and emotion of the song, the more likely it will be that viewers will want to watch your anime music video more than once.
After you finish your storyboard and have determined what footage you need, the next step is to record the footage you need, or rip the footage you need off of your anime DVDs.