You were willing to betray your captain to save your people; I sacrificed my body and mind to protect mine. I have the Human's face, but inside I remain Klingon.Voq, to Michael Burnham, after the collapse of his cover as Ash Tyler, 2256.
The surgery involved overlaying the body of a Klingon subject with that of another species, such as a human. The personality of the Klingon subject was buried beneath that of the human subject, with the human being unaware of the submerged personality. The submerged personality would be reawakened when a specific Klingon phrase was spoken to the individual.
The Klingon L'Rell was aware of the procedure, and suggested to Voq that he undergo the procedure, but warned him it would require him to sacrifice everything. After traveling to the House of Mo'Kai, Voq became the first test case for the extreme, agonizing surgery, which transformed him into an artifical hybrid with the appearance and personality of the original Ash Tyler being overlaid on top of Voq's original personality. L'Rell later tried to reawaken Voq, but the procedure went awry, and the two personalities battled it out inside Tyler's body until L'Rell purged Voq's personality from Tyler's brain. Voq apparently died, leaving the original Tyler personality in charge of his body again, but with access to Voq's abilities and memories. (DSC episodes: "Despite Yourself", "The Wolf Inside", "Vaulting Ambition", "The War Without, The War Within", "Will You Take My Hand?")
In the 2260s the Imperial Intelligence operative Gralmek underwent a form of choH'a' which gave him the appearance of a human named Arne Darvin, however there was no personality reprogramming in this case. (TOS comic: "Beneath the Skin")