Study current music trends and attempt to emulate modern styles in your songwriting. Record companies want songs with good hooks and catchy melodies.
Research record companies you might be interested in writing songs for. Song-pitching companies and books like the "Songwriter's Market" series offer databases of record companies and the types of songs they are looking for.
Record the most professional-sounding demo you can afford, as close to the proposed finished product as possible. If you don't have the funds to record a full-blown demo, record a vocal/guitar or vocal/piano demo that showcases the melody and lyrics of your song.
Contact a music lawyer or record producer. Many record companies, especially major labels, don't accept unsolicited demos. It helps to have professional representation when approaching a record company with your demo.
Submit your demo along with a letter of intent to the record company you want to write songs for. Reference your professional songwriting experience, if you have any, and describe how your songs might fit with the record company and/or for one of its artists.
Negotiate a contract with the record company. For example, you could work on a song-by-song basis or as a staff songwriter, living on advances from the record company and submitting all of your songs you write to them.