You break into television editing by a combination of two ways in my experience:
1) Know someone. You need to get out there and network, do docs, do favors for everyone you can, make friends.
2) Be tenacious. Never stop, if you get an opportunity, grab it and run with it. I had a friend who worked for CBS and he got a good reputation by editing on two machines when he was an assistant night editor. While one was capturing logged footage he’d go to the next available Avid and log some more while other guys would surf the web. Be the former guy and not the latter guy.