The Constitution was rather vague on the role of the Vice President. The only role that was defined was for the vice president to preside over the legislature and be a tie breaking vote in the senate. For this reason both he and Washington believed the vice president was part of the legislative branch of government.
That said, here is how Adams defined the role of Vice President.
1. One of the first arguments before the new legislature was to determine how to address the new president. Believing he was part of the legislature, Adams participated in the debates. He said he believed the president should be addressed as "Your Majesty." Yet he was mocked and ridiculed by senators, who called him "His Rotundity" and "Duke of Braintree." Senators then decided to shut him up by ruling that he could not participate in the debates. Since then, the only role of the vice president in the Senate is to cast a tie breaking vote. While Vice President, Adams used his pen to break 31 ties in the senate
2. His most important tie involved his confirming Washington's ability to remove officials he appointed. This blocked attempts to weaken the power of the president. He also decided on a key argument between the north and the south. Northerners wanted the U.S. government to pay war debts incurred by the states during the war. Southerners wanted the U.S. capital to be in the south.
3. Another tie breaking vote determined how war debts would be paid and where the new capital would be built. Northerners wanted the federal government to pay war debts incurred by the states. Southerners wanted the new capital to be in the south. Hamilton and Jefferson agreed to a compromise whereby southerners would agree for the federal government to pay off war debts, and northerners would agree to vote for the capital to be on the border between a northern and a southern state. Adams cast the tie breaking vote in favor of this. Maryland and Virginia agreed to agreed to contribute land for a new District of Columbia.
5. Because Washington and Adams believed the vice president was part of the legislature, the two rarely spoke to each other during their terms. They did this because they did not want to violate separation of powers.
6. Because the senate only met a few months of the year, and because the president had few powers, John Adams was bored with the job. During his second term he would often travel back to Massachusetts. This enticed Washington to quip: "Presuming that the vice president will have left the seat of the government for Boston, I have not requested his opinion to be taken."