The system we have supports two major parties. That's it. Once in a century or so, a third party rises up far enough to be noticed by the public. They're either absorbed into one of the two parties, take over one of the two parties, or dies off, leaving just two parties. It's always been that way in this country, and always will be so long as we have the current Constitution. The Constitution could be amended to put in place a proportional federal legislature *, similar to what Australia or Germany has, but until we do so, our system is only going to support two parties. And you can bet your nipples that both parties are going to fight tooth and claw to keep the Constitution from being amended to allow for a proportional legislature, because it would hurt the power of both parties.
It could still get done. It's very unlikely to happen without a successful revolution and new constitution, but in theory, it could be done. First, you need to start getting states to switch over to a proportional legislature by amending their state constitutions. You could conceivably get enough voters in a state to make that switch, particularly in states that historically have had a lot of nasty partisanship and in fighting in the state legislatures. You have to focus on people's dissatisfaction with their state politics. Once there's a few states with proportional legislatures, as long as nothing really big and bad happened in any of those states, it would be a matter of slowly taking over new states, state by state. If we could get enough states that thought that having a federal proportional legislature was a good idea (at least 38 states), then the states could offer up an amendment (this is constitutional, but it's never been done... every amendment has come from Congress).
This isn't necessarily the only way to break out of the lock the two party system has in this country, but it's the only one I'm familiar with. It's also a pretty popular form of legislature among first world democracies.
* A simple proportional system works like this: When you go to the polls, you vote the for the party you want. In this election, I'd vote Democrat. Someone else might vote Republican. You might vote Green. When the votes are tallied up, it might look something like this: 45% Democrat, 22% Republican, 21% Tea Party, 10% Green, 1% Socialist, 1% Constitution. If this was a vote for Senate that we were making, we'd probably end up with 45 Democrats, 22 Republicans, 21 Tea Partiers, 10 Greenies, 1 Socialist, and 1 Constitutionalist, and those would be our Senators for the next 4 years. There are many, many different variations on the general idea of a proportional system... the example I listed is probably the simplest variety.