Leslie Odom Jr. was leaving nothing to chance.

It was the fall of 2013, and Lin-Manuel Miranda had emailed, inviting him to an early workshop of the show that would become “Hamilton.” Just an informal gathering — a bunch of actors around a table at a rehearsal studio, reading aloud for the first time from the musical’s second act. Mr. Odom could play Aaron Burr.

It had been only a few years since he shot a TV pilot for a show that did not get picked up, leaving him so disheartened about acting that he started applying for jobs as a hotel clerk. But he had been moved to tears as an audience member at a reading of the first act earlier that summer at Vassar’s Powerhouse Theater.

So he leapt into action. He learned Burr’s big songs. He read up on Hamilton. He dressed to kill.

“You have to walk toward the things that make you alive,” he said.

The show’s director, Thomas Kail, took note. “He was coming to play,” he said. “He was inspired.” But, just in case, Mr. Odom aggressively pursued Mr. Kail as “Hamilton” progressed. “I didn’t hide any of my cards,” he said. “I would text Tommy after every single reading: ‘I love this part.’ ‘I want to stay with this part.’”