Study the artists you'd like to write for. Write songs you believe they would like to record, but do more than that. Examine an artist's CDs. Look for information about who manages him, what record company he records for, and any other information that will give you somewhere to send your songs.
Record a demo of your song. Invest in a studio session and get the best recording of the demo you can. Try to get a singer who sounds similar to the artist you have in mind. You want the artist and her manager to be able to imagine the artist performing the song.
Purchase a copy of the "Songwriter's Market." This is an essential guide to finding managers, producers and record companies looking for good songs. Chances are good you'll find a way to get your songs into the hands of a recording artist with this book. It contains complete contact and submission information for hundreds of music industry professionals.
Submit your demo, a brief bio and a short query letter introducing yourself to the contact you select. Give a little bit of information about the song, state why you feel it is perfect for the artist in question, and close by thanking the contact for his time.
Continue to write and submit songs to various producers, artists and record companies. One song may not be enough to get your foot in the door, but persistence and talent together can eventually pay off with a successful songwriting career.