Use the Internet.
The least expensive--and most difficult--way to get on TV is to gain some kind of notoriety; the internet is a perfect tool for that, as people like Gary Brolsmi of Numa Numa Dance fame have proven. For a short time you couldn't turn on the news without seeing his face numa-numa-iei-ing across the screen to the tune of that insipid and insidiously catchy song. The sheer oddity of the video made it spread through the internet like wildfire, before gaining the attention of the news media. Trust me, I was surprised as anyone else to turn on CNN and see the Numa Numa Dance.
But that's only one example out of thousands of videos on the 'net that have failed to gain notice or fame. The popularity of your animation catching on as a meme is dubious; the internet is a fickle place, and even if you gain notice it could take more months or years than you're willing to wait for it to gain enough momentum and discussion to be noticed by the broadcast media and scooped up.
In addition, that sort of fame is like fireworks: a quick burst of bright light, and then it fizzles out and you're forgotten after a few months. So while you should still use the internet to promote yourself by posting on review/collection site and on your own website, if you're looking for just a simple broadcast airing of your program/animation you may want to try more conventional methods with more surefire results.
Submit your work to film festivals and competitions.
There are dozens of film festivals, conventions and competitions across the nation annually, with Sundance being only one of the most famous. If you want hundreds to view your animation while you gain publicity and recognition, this is one of the most surefire ways to do it; it's like using the internet, in a way, but taking more of an "inside track" with more opportunity for people who actually work in the film, media, and broadcast television industry to see your work. Every year we hear about some new filmmaker who made a huge splash at Sundance and then moved on to be shown on television or even on the big screen; if you give it a try, next year we could be hearing about you.
Contact television stations.
ABC, NBC, Fox, the WB--every TV station has a marketing department and a suggestion department, and it's not that hard to find a phone number or e-mail address on their websites to contact them. If you have an idea, put together a proposal, make a tape, and send it in. Pitch the idea--but be aware that you're going to get rejected, or ignored, again and again and again.
There are dozens of broadcast stations and hundreds of cable stations; if you have to, start off small, with educational stations like PBS or locally-operated cable channels with a smaller viewer base and wide range of programs. Depending on your idea and the quality of the work that you send in, you may or may not be able to secure a little air time without paying for it, as your animation piece is featured as part of a news broadcast or on some other show controlled by the station. Unfortunately, the odds of your idea gaining any attention through this method are very low.
Of course, if this fails, then there's only one way left to go...