It has often been argued that the Founding Fathers were against political parties. Some of them undoubtedly were, but others of them founded political parties. These included John Adams and Alexander Hamilton (founders of the Federalist Party), and Thomas Jefferson and James Madison (founders of the Democratic-Republican Party). They weren't always called political parties - often they would use less controversial language like "the political friends of Mr. Hamilton" or "the political friends of Mr. Jefferson" - but they were parties in every sense of the word.
Critics of political parties make much out of George Washington's opposition to them. But it's easy to oppose political parties when your self-interest doesn't require their support, and George Washington is the only presidential candidate who was ever elected without the support of a political party. His reputation for walking away from power, along with his remarkable war record, made it so he didn't need parties. All he had to do was not say he wouldn't be president, and he would be elected. Most of the other founders, by contrast, did need their support, and actively courted it to gain political office.
Opponents of political parties have often pointed to Madison's words in Federalist No. 10. Some Supreme Court decisions adverse to political parties have quoted this portion of the Federalist Papers. But it is not parties that Madison considers as problematic, but factions; and his definition of the word faction is far different from our definition of political parties. He says in Federalist No. 10 that "By a faction I understand a number of citizens, whether amounting to a majority or minority of the whole, who are united and actuated by some common impulse of passion, or of interest, adverse to the rights of other citizens, or to the permanent and aggregate interests of the community." Thus a political party must be adverse to the rights of other citizens, or to the "permanent and aggregate interests" of the community, to qualify for his definition of faction; and it would be a gross overstatement to say that all political parties meet this definition. I certainly believe that some parties do (notably Democrats), but to say that all of them do requires cynicism of a rather extreme nature.
Madison did argue in Federalist No. 10 that majorities of the same party made it easier to "concert and execute their plans of oppression," but said soon after in that paper that if you have "a greater variety of parties and interests, you make it less probable that a majority of the whole will have a common motive to invade the rights of other citizens; or if such a common motive exists; it will be more difficult for all who feel it to discover their own strength, and to act in unison with each other." Thus it is not the existence of parties that he sees as the problem, for he says that a greater variety of them removes the problems he identifies with faction, but it is any party gaining a majority that he sees as the problem.
And as noted before, Madison must not have been against political parties all that much if he later helped found one of them (the Democratic-Republican Party), and allowed it to support his successful campaign to become 4th President of the United States. And one other point needs to be made about his opposition to majority parties: he did not feel this opposition throughout his entire life. In the two years prior to his election as president, the Democratic-Republican Party had 80 percent of the House and 80 percent of the Senate, plus the presidency (in the form of his friend Thomas Jefferson), which sounds to me like a majority. (source citation) If he allowed this majority party to support his campaign to become president, he must not have thought majority parties were all that dangerous.
The only solution Madison advocates in Federalist No. 10 for parties gaining a majority is to have a republican form of government, which makes it less likely that a party will gain a majority. (They would have to gain a majority over three separate branches of government, and that is much more difficult.) We have such a republican form of government, and we are thus already implementing the one solution he advocates for this problem in Federalist No. 10. No additional solutions can be advocated on the basis of this essay, let alone on the basis of Madison's actual behavior with regard to political parties (which was decidedly pro-party). Opponents of political parties will have to turn elsewhere to support their position, because most of the major founders disagree with them. The American tradition of supporting political parties goes back to their time, and political parties in America were invented by them.
Some thoughts about third parties
What are the "Federalist Papers"?
Other posts about the Founding Fathers