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Really There is nothing at the Constitutional level. The opinion of the Founding Fathers seemed to be that of Thomas Jefferson: political parties are a necessary evil. Hamilton, Adams, Jay, Marshall were the big proponents of a strong central government which they pushed in the Federalist Papers, and became the leaders of the de facto Federalist Party that was initially in control. Washington was a Federalist as well, but attempted to be what we would call bi-partisan today. Jefferson and Madison were the founders of the Democratic-Republican Party that argued for stronger state governments/rights.
Over the years a complex web of Federal and state laws have emerged around party organizational structures, donor identification, campaign financing, etc. Although the biggest effort to make these transparent and limit the ability of billionaire and corporate donors to dominate campaign financing, passed by Congress, was ruled un-Constitutional under the First Amendment by SCOTUS in a decision based on the assumption that corporations have the same free speech rights as individuals. Something that has to have the Boston Tea Partiers turning over in their graves.
So the initial issue for any new political party is getting on the ballot, and as a previous poster stated, the ballots are controlled by each state and the rules, steps, vary with each. So it is easier, in most case, to take over one of the smaller parties already on the ballot in as many states as possible and contrary to general belief, we have a multitude of "third parties" -- just not one with sufficient clout to matter much (with the exception of the Presidential runs by Ross Perot and Ralph Nader) where it could be argued both played a spoiler roll. Part of the battle over voting rights currently in the House and Senate is over this hybrid model where the States control the ballot and voting, even in Federal general elections. So you have the Democrats pushing for a more Federalist expansion of voting rights, and the Republicans trying to rigidly control voting in the Red states.
The major issue, though, is access to a large enough megaphone to become a big enough voice to be heard and that the two existing parties have a de facto choke hold on by controlling the rules on who participates in broadcast debates, keeping the price of campaigning extremely high, etc. Ross Perot and Ralph Nader broke through by virtue of their name recognition, and in the case of Perot, his personal finances.
