If we were to only look at on-screen examples as canonical references, perhaps one of the earliest episodes involving Klingons gives us a good indication (and it appears has not been contradicted or specifically stated to have changed all that much) -- that would be ST:TOS Errand of Mercy.
The Klingons set out to conquer and incorporate Organia. When Kor beamed down their troops, they started to set up an occupation. Kor told the Organians that their way of life would be changing, they would be serving the Empire -- not that they would all be massacred in a mass genocide. In essence, their culture would be adapted to service the Empire, and resistance was futile (punishable by death if they fought against the Klingon forces). As long as the Organians submitted and complied, they would continue to exist on Organia. Would we then see Organians serving as officers and crew aboard Klingon Defense Force vessels, and interacting with (facing off against) the Federation vessels that the shows invariably followed? Probably not. They would largely be kept on Organia.
If we extend the view to include "soft canon" novels and the canon of the Klingon language, tlhIngan Hol, we see a more detailed view. If a race is conquered (and presumably determined to be either less than honorable or incapable of induction into the warrior caste) then they become jeghpu'wI'. -- "conquered people" ["One(s)" who has/have surrendered"]. They are more than slaves, but less than citizens. They are the lowest rung of the societal ladder, relegated to the most menial and undesirable tasks and functions that Klingons of higher castes feel is beneath them. Generally they are expected to perform their function and ignored. While certain unruly or dishonorable Klingons may bully or even be violent to jeghpu'wI' who fail in their function, jeghpu'wI' are typically not routinely beaten or abused as some Earth cultures of the past have treated enslaved races. (Perhaps owing to the Klingon proverb, pujwI' HIvlu'chugh quvbe'lu', "There is no honor in attacking the weak." [The Klingon Way, p.24])
In the I.K.S. Gorkon novel trilogy, we see examples of a jeghpu'wI' character being "lucky" enough to serve aboard a Klingon Defense Force vessel as cleaning staff in the mess hall. The overwhelming majority of the ship's crew typically completely ignores and disregards the presence of the jeghpu'wI' character. Would this character be seen by a Starfleet officer in a typical interaction with the Klingon vessel? No. They would perform their function in the mess hall, then return to their "servant's quarters" when not on duty. So races that were deemed weak of body and spirit would be subjugated into the role of jeghpu'wI', allowed to live so long as they served the Empire.
Are all conquered races destined to serve in this role, this "lowest of low" caste? Perhaps not. An interesting example is the Children of San-Tarah from the I.K.S. Gorkon novels. The Klingons view this primitive caninoid race as fellow warriors, rivaling the Klingons in fierceness, prowess and honor. Captain Klag determines the Children of San-Tarah are worthy foes, gives his word of honor to them, and battles other Klingons to uphold that word.
In the end, the Children of San-Tarah are incorporated into the Klingon Empire, essentially submitting and agreeing to be incorporated. Chancellor Martok and Captain Klag feel that while the Klingons have much to teach the Children of San-Tarah, the Children of San-Tarah also have much to teach the Klingons in return.
So it would seem that the Children of San-Tarah would have been allowed to join the warrior caste, perhaps even serve aboard a Defense Force vessel as a crew member or even officer; not relegated to simple and menial tasks such as janitorial services. Would they, then, technically be jeghpu'wI'? Perhaps certain individual San-Tarah would be kept in jeghpu'wI' caste while other individual San-Tarah would be allowed to elevate into the farmer or warrior caste.
Another example of non-canon, in the MMORPG computer game Star Trek Online, the Klingon Empire has conquered and incorporated the Gorn Hegemony, Orion and Nausicaan empires, and the Lethians; however, all of these races are allowed to enroll in the Klingon Academy and serve as full crew and officers in the Klingon Defense Force. This may make sense if the Klingon High Council had determined that these races were innately capable of becoming honorable warriors.
Taking all of this into consideration, we can see that while the Klingon Empire would seem largely dominated by and apparently made up almost entirely of full-blooded Klingons, there would be colony worlds on which various alien species were conquered and subjugated to serve the Empire (although Federation and Starfleet vessels may seldom, if ever, encounter them as they would be deep inside Klingon territory).