How did George Washington carry out the role of "Chief of Party" as president?
In modern US Government textbooks, "Chief of Party" is typically given as one of the "hats" that a president wears. However, I would say that George Washington largely did not carry out this role. That is because there really were no political parties as such when Washington was president. We would say Pres. Obama is the head of the Democratic Party because one of his jobs is to lead his party and to try to get it to win in its competition with the Republicans. But Washington really did not have a similar situation.
When Washington was president, there were no official political parties. In fact, members of his cabinet (Jefferson and Hamilton) would later form the first two parties in US history. This shows that Washington was clearly not acting as party leader -- he had people who would later be from different parties right there on his own team.
It was only after Washington left offices that parties formed so could not act as a Chief of Party the way that modern presidents do.