1) A Movement Within One of the Major Parties
The right-wing takeover of the Republican Party that began with Ronald Reagan is a good model. Methodically and relentlessly, the ultra conservatives plotted out a slow conquest of the GOP that took 20 years to come to fruition in the presidency of George W. Bush.
The genius of this intra-party revolution was the coalescing of religious, economic, and military zealots. Individually, they posed no threat to the moderate wing of the Republican Party. Together, they were able to shift the entire nation several notches to the right.
Some progressives are hoping for this kind of process to happen in the Democratic Party, shifting the nation back to where it was in 1980. There are currently a number of organized effotts to groom progressives for movement into higher offices as Democrats.
I'm personally skeptical of such a movement finding success on the left, as these candidates are often required to sell their political souls either to get elected, or to have influence and power once they get into office.
2) A True Third Party Movement
Historical precedents are rare for the building of a new party from scratch, but it has happened, sort of. The Republican Party has a complicated early history that involved several new parties emerging as the Whigs declined and abolition became a primary issue. Nonetheless, that era does prove that the two major parties are not indestructible, and that given the right circumstances, a new party can arise.
What might be impossible, I'm told, is for a third party to arise while the other two remain strong. Invariably, the old party that is closest in ideology to the new one will weaken and die.This is why many Democrats do not want to discuss the possibility of a progressive third party. They fear it will either play spoiler, like Nader in 2000; or will grow strong enough to weaken the Democrats and give power over to the the Republicans.
The true third party method, therefore, is a catch-22 - unless one is so disenchanted with the two-party system that they don't think it matters which of the old parties is in power.
3) An Independent Movement That Overlaps With A Major Party
The obvious example here is the Tea Party. Focused sharply on economic issues like the national debt, Congressional spending, and taxation, the Tea Party became enough of a force to spook the Republicans into action. Quickly co-opting the Tea Party message and many of its candidates, the GOP shifted even further to the right as it worked not to see its base fractured.
The Republicans were successful at neutralizing the threat, but the ongoing influence of the Tea Party is indisputable, from the GOP landslide in the 2010 mid-terms to the dramtic shift to the right in Congress' economic agenda.
To me, this seems like the best model for activists on the right or the left to use. An independent, grassroots movement that unabashedly promotes a specific set of values can gain considerable steam if it can find a way to unify a diverse set of followers. And as a voice that is independent of the maor parties, it would have the freedom and the courage to truly tell it like it is.