Commander Kruge was the commanding officer of a Klingon Bird-of-Prey in the late 23rd century. In that capacity, he recruited his lover Valkris to obtain intelligence on the Genesis Device, a Federation technology designed to instantly terraform an entire planet.
After obtaining the information and destroying the vessel Valkris was aboard in order to preserve secrecy, he ordered a course set for the Genesis Planet, on a mission to subvert the technology from the Federation and utilize it as a weapon. Arriving at the planet, they encountered the USS Grissom, a Federation science vessel assigned to study the newly-formed world. In an attempt to gain hostages with which to bargain for the Genesis technology, Kruge ordered his gunner to target the Grissom's engines in order to disable the vessel. However, in what the gunner described as a "lucky shot," the Grissom was destroyed, an event to which Kruge responded by instantly killing his officer. Kruge later discovered, however, that a landing party from the Grissom had survived on the planet's surface, and subsequently led a team to search for them.
Kruge and his officers eventually located the team, consisting of Starfleet officer Lieutenant Saavik, son of Admiral James T. Kirk David Marcus, and a rejuvenated Captain Spock, whose body had been revived by the Genesis wave. Taking the team as hostages, Kruge demanded to know the secret of Genesis, and refused to believe Saavik when she informed him that the technology was fundamentally flawed, and that the planet they were on was on the verge of destroying itself. Upon the arrival of Admiral Kirk and the USS Enterprise, Kruge returned to his ship and attempted to ambush the Federation ship. The Enterprise crew detected the Bird of Prey's cloaked approach and got off the first shot. Kruge returned fire, expecting to be destroyed because the Federation ship outgunned him ten to one. Unknown to Kruge, the Enterprise was crewed only by Kirk and a handful of bridge officers. Without its normal crew of hundreds, the Enterprise was effectively disabled when Kruge's torpedo knocked out its automation center. Kirk, bluffing, opened communications to demand Kruge's surrender. Sensing that Kirk was hiding something, Kruge instead ordered Kirk's surrender, threatening to execute the prisoners as "enemies of galactic peace." As proof of his commitment, Kruge ordered his men on the surface to choose at random and kill one of the prisoners. As one of Kruge's men moved to stab Saavik, David intervened, sacrificing his own life to save her. In response, Kirk deceived Kruge into believing that he was indeed surrendering, allowing a Klingon team aboard the Enterprise while he and his own crew surreptitiously set the ship's auto-destruct and beamed down to the planet.
After the destruction of the Enterprise and the death of the bulk of Kruge's crew, Kirk contacted him from the surface, demanding to be beamed up. Kruge, however, chose to beam down himself to confront Kirk, allowing the rest of his crew to transport to the Bird-of-Prey. Engaging in a fight with Kirk amid the conflagration of the dying Genesis planet, Kruge lost his footing as the ground gave way beneath him. Kirk caught Kruge as he fell, but instead of accepting Kirk's offer of mercy, he attempted to yank them both to their deaths. He failed, and was left dangling from Kirk's leg over an immense lava flow. Kirk then kicked him off the cliff where he fell to his death. (Star Trek III: The Search for Spock)
Background information Edit
Kruge was played by actor Christopher Lloyd (although Edward James Olmos of Battlestar Galactica fame had also auditioned for the role).  According to the script, Kruge was described as a "Battle Commander" and was "a Klingon War Lord of handsome but frightening presence, and relative youth." This explains why Saavik, Valkris and his crew referred to him as "my Lord." This form of address among Klingons is never seen again.
The stop-motion puppet of Kruge's fall from the precipice following his defeat with Kirk, was animated by Tom St. Armand and Dennis Muren. The scene was shown during the video montage at the end of the Star Trek 25th Anniversary Special, followed by the text "He Boldly Went Where No One Went Before."
Describing the role, years later, Lloyd said, "To this day, I don’t know why they cast me, because I hadn’t done anything that I can recall that bore any resemblance to the character of Captain Kruge. They just played a hunch, I guess. And I loved doing that. I mean, he epitomizes somebody with absolutely no moral conscience. He even blows up his so-called girlfriend in another spaceship. They have a short conversation at the beginning, and he doesn’t even apologize. She’s amenable because… well, it’s for whatever political reasons. But, yeah, he’s just evil[...] He’s demonic. There’s no conscience in place at any point, and he has no apologies for any of his actions. He just goes out and destroys and kills and creates havoc until he gets what he wants. And that was fun to play. I loved all the makeup and the clothes, the whole Klingon look. It was a joy. And Leonard Nimoy directed!"
A Klingon named Kruge appears in the novel Faces of Fire by Michael Jan Friedman, which takes place several years before the events of Star Trek III: The Search for Spock. It is unclear whether he is the same character as appears in the movie, but the two are strongly similar.