Determine the owner of the movie you wish to license. The owner of the movie's rights is often the distributor of the film or the major studio that produced it. Oftentimes, studios sell their films for after-theater distribution. Check imdb.com for the distributor.
Find the film owner's clip licensing office. All major studios have either an individual or an entire office devoted to licensing film and television clips to individuals and organizations. Check out the studio's homepage or call the studio's switchboard to find the best way to contact the clip licensing office.
Contact the clip licensing office. When calling, explain the context in which you wish to use the film clip. The office should ask you either to fill out a form or to send an e-mail message describing details about the project and the clip you wish to use. The office will also ask you the length of time you wish to license the clip for and if you will require a "dub" (media jargon for a copy) of the clip.
Await a price quote. Within a week or so, the clip office should send you back a quote to use a clip as well as a term sheet with guidelines for using the clip. It's not uncommon for a clip office will deny a request outright. Wait at least a week before contacting the clip office, should you not receive a response. These offices are swamped with requests.