Bonnanno conspired with Joe Magliocco, successor to Bonanno's ally Joe Profaci, to whack Gambino because Carlo tried to keep Magliocco off the Commission and was actively lobbying people in Magliocco's family to depose him. Lucchese was Gambino's ally. It was a twofold power play: first, to keep Bonanno's ally Magliocco on the Commission so they could have an old-time Sicilian faction (they called themselves "Men of Tradition") to put up against the others. Second, to stop Gambino's ascendancy to leadership of the Commission. If Bonanno had succeeded, he would have assumed that he was the top dog. But, he would have had to worry less about the Genoveses and more about his cousin, Stefano Magaddino, the Don of Buffalo NY, who was on the Commission at the time. Magaddino was angry with Bonanno for allegedly poaching his territories in eastern Canada.

As you probably know, Revis: The plot failed because Joe Colombo, then a capo in Magliocco's family, ratted it out to Gambino. The Commission ordered Magliocco to step down in Colombo's favor and fined him $55k. They ordered Bonanno to come before them and "explain himself." When he refused, they ruled that he was no longer Don, and put Gaspar DiGregorio (who was best man at Bonanno's wedding) in his place. Magaddino led that move--DiGrigorio was his brother in law. As you also probably know, Bonanno claimed in his autobiography that Magaddino arranged his "kidnapping" in 1964. (I believe Bonanno had himself "kidnapped" to get out of the line of fire and to duck a grand jury subpoena.)