Thanks to our voting system of single-member district, first-past-the-post elections, structural forces pretty much guarantee that American elections will be between two major political parties, no more. (To political wonks, this is called "Duverger's Law.") There are ways to change the system to make third parties more viable—my favorite is called Instant-Runoff Voting—but for the most part, we can neglect those possibilities, since the two major parties are hardly interested in damaging their stranglehold on power.
Still, third parties exist. The most influential are the Libertarian and the Green Parties; and every so often people in the Republican Party rumble about breaking off and forming a "real" conservative party, perhaps organized around the Tea Party. Before anyone takes such a step, it is worth reviewing what exactly a third party is intended to do.
Third parties are generally incapable of winning elections. The exceptions, apparently, are either when the candidates of both major parties spectacularly self-destruct at the same time (as when Jesse Ventura won the governorship of Minnesota), or when the race is for a local position in a low-turnout, low-interest race that can be swung by a core of dedicated partisans. If you hope to win on a third-party line, my advice would be to find a boring political office in a sleepy jurisdiction where the incumbent is annoying his constituents and you have an unusual number of supporters, build a disciplined organization of volunteers, knock on a ton of doors, make friends with all the local power centers you can, and generally fight your way to victory by being better at retail politics than the other guys.
This seems a tall order, given that third-party activists are generally attracted to ideological conflicts and not retail politics. But the key, it seems, is to avoid drawing too much attention to yourself, so as not to spook the national parties into bringing their resources to bear. Done right, this strategy can help a third party develop a bench of seasoned public servants who are well-placed to advance to the higher ranks in politics.
What it won't do, however, is to drastically change people's minds on the major ideological conflicts of the day, the ones that motivate people to join third parties in the first place. Which is why most third-party candidates run for offices they cannot possibly hope to win: the presidency, governorships, congressional seats. That a handful of libertarians and socialists have managed to win elections to Congress over several decades proves the point: all the other candidates were not running to win.
So why run? The general theory is that by using the election as a platform to argue your point of view, you can influence people to shift their views over time. Third parties can serve as incubators for ideas that then diffuse into one or both of the major parties, carried there by sympathizers who then infiltrate the establishment. Libertarianism has had some success in influencing the Republicans in this manner, as the Greens and Progressives have to the Democrats. Still, it seems to me that most of the work there has been done by think tanks and public intellectuals, and the radicalized volunteers for third-party campaigns—not by third-party candidates themselves.
Most of those candidates have been quite counterproductive, in fact, in getting their real message out. Think of Alan Keyes's quixotic 2004 campaign against Barack Obama for the Illinois Senate seat. How many people did he convince that libertarianism is a reasonable ideology, compared to the people convinced that he was a machine-gun-crazed lunatic? (Granted, Keyes was nominally Republican, but let's be honest here.)
So if you are an activist for a third party and you want to run a campaign for the purpose, not necessarily of winning, but of changing the public mindset to favor your cause, how should you go about it?
It seems to me that the best tack is to eschew the traditional playbook for running an election. You are not going to win the vote; to try is to play into your opponents' strengths. What you want is not to assemble 50% of relatively ill-informed voters living in an arbitrary patch of land; you want to get into the head of the literate, engaged, politically aware people all across the country. The model to follow then is not retail politics, and not the sad pathetic strategy where you boldly state your positions in the most uncompromising language possible, and then sit back and wait for the public to spontaneously have a road-to-Damascus moment. You need a way to get people's attention, to break them out of the traditional patterns of political thinking, all across the country. You are trying to blow their minds.
In short, third-party activists ought to be developing a model of the political campaign as performance art.
Ditch the comprehensive platform. Ditch the exhaustive philosophical pamphlets; plenty of time for people to read them once you have already piqued their interest. Instead, create visceral, stunning moments of experiential clarity showing where your model is superior to that of the major parties. Create viral videos; file high-profile lawsuits against the corrupt parasites who exemplify the worst ills of the system you mean to fight. Gather volunteers to clean up a slum or build a house or plant a farm, organized around the principles you espouse. Distill your philosophy down to its very essence, an essence that explodes all the assumptions about politics, and then repeat that essence constantly—but not as an explicit lecture. Instead, make it the theme of every image, every piece of music, every organized gathering you do.
One example where people almost, but not quite, did this right was the Occupy Movement. The Occupy activists managed to create a massive event and transfix the attention of a national audience for a short time. If not for the general incoherence of their message, their ideals, their program, and generally their plan for getting anything done at all (including maintaining order in their own encampments!), the Occupy crowd could have pulled off a massive shift in our politics. As it is, even with the general failures of the movement, they still managed to galvanize particular covert tendencies within the Democratic party into coming out into the open. (Not that this was a good thing, but that's not my point.)
So yes. Politics as performance art. That seems to be the way to change things in this country.