Research the musician. Paul New, a drummer from Wisconsin who calls himself "The Wisconsin Music Man," says he favors interviewers who do solid research before the interview. Investigate the musician's past projects. Listen to his albums. Go to a show. Or, if possible and reasonable, interview the musician's friends and family first. These efforts will give you the right fuel for a quality interview.
Contact the right person. If the musician is famous, you'll most likely have to go through a manager or publicist. Check the liner notes of a recent album and look for names, whether of people or a management company. If less famous, though, a quick scan of the musician's website can get you an email address or other contact information that may lead to an interview.
Choose the right setting. New mentions one time when a journalist requested an interview at a bar on a Sunday morning at 9 a.m. He, a bit confounded, was not in the best mood for the interview. Choose a convenient place that will make the musician comfortable, even at his own home, if he prefers. Or, go out to eat and make your interview a casual conversation over dinner.
Ask the right questions. Strive to make the musician think by asking questions he's not accustomed to. Avoid fawning over how much an album meant to you or asking about his musical influences or what the musician's "sound" is. These are common questions he might hear more than he would like, or are questions to which you should already know the answers from your research.
Remember to enjoy it. It cannot be overstated that the best interviews happen when both sides are relaxed -- so that there aren't really sides at all. Ideally, it should be a conversation. Stay focused, but loosen up. There's a reason why musicians have a reputation for being casual and carefree. Match that vibe and your interview will be all the better for it.