A good option to consider when trying to 'break in' to composing for TV is writing for one of the well known music libraries. While this option is far from your ultimate goal of composing directly to picture it will provide you the opportunity to get some of your music placed in Film, TV, Web and various other types of production.
Once you're able to secure some placements through a music library you'll:
a.) have made a few $ (which we all need!).
b.) added some relevant experience to your resume.
c.) hopefully have made some good contacts at the library who can maybe help you with advice on moving toward your goal of composing.
Stick with it, be open to trying new things and above all keep a positive attitude...media production clients generally prefer to work with composers that are easy to communicate and get along with! (hint: don't be a diva until you've scored a few major motion pictures)