Contact the American Society of Composers, Authors, and Publishers (ASCAP). ASCAP administers the rights to more than 8.5 million songs, including the majority of songs released by major labels. You can search on ASCAP's website to see if they administer the song. If so, you can select the necessary license holder from their website and contact them to purchase the license. If the song is not administered by ASCAP, you can try the other major music rights administrator, Broadcast Music Incorporated (BMI). You can also call either company; both companies are set up to provide this information to you. SESAC is the third largest music administrative company in the US. The rights organization for which the song is registered is also listed in the credits on every CD.
Find the publishing information of the song you want to use. This will be on the album if you own that, usually at the end of the credits for each song. If not, the song's information should be available on the Internet. You're looking for the specific company that owns the publishing rights to the song, which in the case of independent albums is usually the musician who recorded the album.
Contact the owner of the song directly. Use Internet searches to find up-to-the-minute contact information, as the address on your album may be out of date. If the publishing company is out of business, contact the musician directly. Musicians on smaller or independent labels will often be accessible through their website or other contact information. You can broker with them directly the rights you need for your work.