The Empire was founded by Kahless the Unforgettable after he united the Klingon homeworld of Qo'noS in the 9th century. After the Klingons took to the stars they formed an Empire as they colonised new worlds and conquered other species.
During the 23rd century, the Klingon Empire and the United Federation of Planets had a extended period of cold war. This finally erupted into a brief war in 2267, which was stopped by the intervention of the Organians who forced both sides to make peace. (TOS: "Errand of Mercy")
In 2293, the destruction of the moon Praxis and the economic strain placed on the Empire by their cold war with the Federation saw the Klingon Chancellor, Gorkon, propose a controversial peace treaty with the Federation. Despite efforts on both side to derail the process, which included the assassination of Gorkon, the historic Khitomer Accords were signed formalising peace between the Klingons and the Federation. (Star Trek VI: The Undiscovered Country)
By the 2340s, relations with the Federation had deteriorated to the point of near war. However, the sacrifice of the USS Enterprise-C to protect the Klingon world of Narendra III from a Romulan attack saw relations improve.(TNG: "Yesterday's Enterprise")
In 2372, the Klingon Empire went to war with the Cardassian Union after they came to believe that Changelings had infiltrated the Union's government that saw the Detapa Council overthrow the Cardassian Central Command. The Federation Council denounced the invasion and in response Chancellor Gowron took the Empire out of the Khitomer Accords, ending their alliance with the Federation.
This also nearly led to war with the Federation after the USS Defiant rescued the Detapa Council and it exchanged fire with Klingon warships and the Klingon fleet attacked the Federation-administered Deep Space 9. Faced with a war on two fronts when Federation reinforcements arrived, Gowron pulled his forces back and fortified the Klingons new holdings they had captured. (DS9: "The Way of the Warrior")
In late-2372, the Klingons demanded the Federation leave the Archanis sector, despite ceding it a century previous. When the Federation refused to the Klingons declared war on the Federation. After three months General Martok was exposed as a Changeling who had been manipulating the Klingon Empire and a ceasefire was agreed to. (DS9: "Broken Link", "Apocalypse Rising")
Several weeks later, the Empire briefly broke the ceasefire when at least four battle groups reengaged the Federation forces along the Federation-Klingon border. After a brief period of fighting the ceasefire was reinstated. (DS9: "Nor the Battle to the Strong")
In mid-2373, the Cardassian Union joined the Dominion and received a large reinforcement of Jem'Hadar warships and troops. They were then able to drive the Klingon Empire from its captured territories. Faced with this new threat Gowron reentered into the alliance with the Federation to oppose the Dominion. (DS9: "By Inferno's Light")
In an alternate universe, the Klingon Empire conquered the Cardassian Union after the Federation descending into a civil war. The Empire then invaded the Demilitarized Zone putting it in conflict with the Maquis. (Star Trek: The Farthest Star: "There's No Angels Here")
The Klingon Empire was a feudal monarchy with an Emperor at its head. However, the true power lay with the Klingon High Council and by the mid-21st century no emperor sat on the throne. In 2369, a clone of Kahless was created and he took the throne largely as a religious figurehead. (TNG: "Rightful Heir")