According to Washington: A Life by Ron Chernow (2011 Pulitzer Prize for Biography), it didn't take very much. I'll get to what little existed, but it's important to know the context of Washington's stature among the United States' leaders and its citizens.

He had served as the President of the Constitutional Convention, which allowed him to remain above the negotiations and wrangling of the delegates, and his quiet authority gave him considerable weight when he did choose a side during debates (he often sided with fellow Virginian James Madison). At the end of the Convention, the ratification process started and this is probably the decisive factor in him becoming president. The Constitution was quickly ratified by five states (Pennsylvania, New Jersey, Delaware, Georgia, and Connecticut), but it was bogged down elsewhere--including in Washington's native Virginia. To this end, Washington did engage in some quiet lobbying with the Virginian opponents of the Constitution, particularly Edmund Randolph. He was also contacted by James Madison who asked Washington to ensure the publication of The Federalist Papers in Virginia (and revealed that he was one of the authors--which is asked Washington to keep secret), which Washington did. This was crucial, since by this time Massachusetts, Maryland, and South Carolina had ratified the Constitution, bringing the total to 8--one state shy of the 9 needed for the Constitution to become law. New Hampshire was the 9th to ratify, but Virginia did as well only four days later without knowing the result in New Hampshire. His public support for the new Constitution was important to a lot of people, both leaders and commoners.

Washington was thrilled that the Constitution had passed, but he was also well aware that he would probably be asked to become president. Many of his peers were already telling him that he was the unifying figure the country needed. Alexander Hamilton in particular, who had served on Washington's staff during the war, prodded his old boss to take up the role if offered (Hamilton also knew that he would likely get a powerful role in a Washington administration). After a lot of prodding, which Washington resisted for a long time, he finally assented to serve if elected. The public had been behind him for a long time, as evidenced by the newspapers of the time. Many of his peers were on board. And it was made all but assured when the electors were to be chosen in January 1789 with the election in February. Even if any of his peers believed in campaigning at the time, no one could have overcome Washington's stature in only a month.

HOWEVER. Yes, there was a noted incident of backroom politics. In the electoral system at the time, each Elector cast two ballots. The first-place winner became president, the second-place person became vice-president. Now, Washington was a symbol of the new Constitution, which was supported by the Federalists and opposed by the Antifederalists. The Federalists feared that the Antifederalists would sabotage the election by withholding votes from Washington, which would allow John Adams (who most people thought would come in second) to win the election. Adams was also a supporter of the Constitution, but he was nowhere near the unifying figure that Washington was and didn't convey the kind of legitimacy to the new system that Washington would. To counter this potential threat, Alexander Hamilton quietly told several electors to withhold votes from Adams so that the possible Antifederalist plot would fail.

In the end, Hamilton's scheme was unnecessary as all 69 electors voted for Washington and Adams came in second with 34. The only lasting effect of Hamilton's backroom lobbying was that it destroyed his relationship with John Adams when Adams later found out about it. Adams considered it a slight that he received less than half of the votes Washington got, so the few votes that Hamilton swayed from him meant a lot to him.

Source: Chernow, Ron, Washington: A Life, parts of chapters "Rising Sun" and "Mounting the Seat", pages 536-551.

I don't think this directly answers your question, but in Decision in Philadelphia there is a short part about how Madison thought one of the hardest parts of the whole debate on the Constitution was about limiting the powers of the Presidency.

Since everyone knew that Washington would be the first executive under the Constitution, it was difficult to reconcile their trust in him with the need to limit what a future President might do. Especially with Washington sitting right there in the room with them while they were debating.

The implication I got from all that was that it didn't require any backroom politics to elect Washington.

In the end, Washington took all 69 electoral votes in the 1788 election.