Burke was part of a conspiracy to assassinate Gorkon, and thus to prevent those peace initiatives from taking place. When the ship carrying Gorkon, Kronos One, was disabled, Yeomans Burke and Samno surreptitiously beamed over wearing environmental suits and gravity boots, and killed several Klingons - including the Chancellor. They beamed back and hid their blood-stained suits in a vent, in the Enterprise mess hall, where they were later discovered by Chief Engineer Scott.
Since the first rule of assassination is to kill the assassins (according to Kirk), Burke and Samno were shot at close range by a phaser set on stun (because a stronger setting would have set off an alarm), and left for dead in an Enterprise corridor. When the senior crew discovered their bodies, they flushed out their killer by announcing - shipwide - that Burke and Samno were in Sickbay, and that they were to give a statement to a court recorder.
It was Lt. Valeris who entered Sickbay to presumably finish the job she started, and she was exposed as one of the conspirators.