Figure out your project's music needs. Which parts of your project need music, and what kind of music will fit best with them? For example, if you're shooting an action movie, a hard rock song might fit. If you're putting together a wedding videography, you'll probably want to go for something lighter.
Look up the information for the publisher of a particular song. You can find such publishing information in the liner notes of a CD.
Visit the website of one of the major performing rights organizations if you can't find publishing information in CD liner notes. The American Society of Composers, Authors and Publishers (ASCAP), Broadcast Music Inc. (BMI) and Society of European Stage Authors & Composers (SESAC) all have searchable databases to look up publishing information on songs.
Contact the U.S. Copyright Office if you're still unable to find publishing information. It also offers an online database on publishing information.
Once you find the publishers of the music you want to use in the background of your project, contact them. Send them a letter stating the intent of your project and the context in which the music will be used. If your project goes against the principles of the publisher or they perceive your project will put their song in a bad light, they might not grant you permission to use it.
Pay the required fee for the use of the publisher's music in your project. They will license the music to you. Use the music in your project, but only within the context in which you stated, or you will be in breach of contract.