Yeoman Burke was a member of the crew of the USS Enterprise NCC-1701-A in 2293. In that year, along with Yeoman Samno, Burke was involved in the conspiracy to sabotage peace talks between the Federation and the Klingon Empire.
Burke and Samno were on duty in the transporter room when Chancellor Gorkon and his party beamed aboard the Enterprise to attend a formal dinner with Captain Kirk and his crew, Burke announcing their arrival with a boatswain's whistle. After the party had left the transporter room, the two traded racist remarks about the Klingons, Burke commenting that "only top-of-the-line models can even talk..."
Following the departure of the Klingon contingent, Gorkon's flagship, Kronos One, was attacked by a cloaked Bird-of-Prey, suffering severe damage. As the disabled Kronos One drifted out-of-control, Burke and Samno beamed aboard wearing EV suits and gravity boots, and began systematically murdering Klingons, including Chancellor Gorkon himself. After returning to the Enterprise, the pair hid their suits in a ventilation shaft in the ship's dining room, where they were later discovered by Montgomery Scott. When it became clear that the Enterprise crew was close to discovering the truth, Burke and Samno were murdered by fellow conspirator, Lieutenant Valeris, via a phaser set to stun fired at close range. Later, in a ruse to discover the other conspirator, Captains Kirk and Spock initiated a sting operation, where misinformation was spread indicating that Burke and Samno had survived, and laid in wait in sickbay to discover Valeris, who had come to finish the job of killing the two yeomen. (Star Trek VI: The Undiscovered Country)