Looking at what happened, it does not appear that international issues were the primary driver.
When the Constitution was first being debated and voted on, there was a group of Anti-Federalists who were very suspicious of it, and in particular the way it concentrated power in the Federal government. This feeling had enough support in the country that a revision was required addressing their concerns (the Bill of Rights), before the document could placate enough of those people to get ratified.
Then, in the very first administration, Washington's treasury secretary Alexander Hamilton introduced a sweeping economic program involving creating a powerful national bank, assuming the previous government's debt, and imposing tariffs to both pay that debt and promote local industry. That may not seem radical to modern eyes, but to the people who were still very wary of a strong central government, that government immediately assuming all that power was alarming.
James Madison and Thomas Jefferson started holding meetings with like-minded people to help organize the opposition to this program in Congress. Hamilton reacted by starting to organize his own supporters under the "Federalist" banner, which caused Madison and Jefferson to need to further organize nationally to compete, and it sort of snowballed from there.
Now to be sure there was a foreign policy dispute between the new parties. For various reasons, after the French Republic was declared the Federalists tended to be much more supportive of the English side in the subsequent wars, while the Democratic-Republicans tended to be more sympathetic to the French. However, that was not what induced the founding of the two parties.