The first race destoyed in the name of the Eternal Doctrine was the Yuptar, a one time ally of the Ur-Quan in the Sentient Milieu. After their loss in the first Doctrinal War to the Ur-Quan Kzer-Za, the Kohr-Ah were banished to circle the galaxy, but were allowed to continue enforcing the Eternal Doctrine.
They became nomads and slowly rebuilt their fleets over the millennia to be equal to, and then stronger than, those of their Kzer-Za brothers. In the mean time, the Kohr-Ah have destroyed countless races. Their ship captains keep relics (usually a single bone of each species they have personally destroyed) as trophies showing their adherence to their race's ideals. Some individual captains have collected thousands of such items.
The Kohr-Ah are also taken with the belief of reincarnation. They believe that by destroying other sentients, cleansing them, they will allow those beings the opportunity to be reborn as Ur-Quan. The Kohr-Ah are thus completely remorseless killers. However, this belief is not necessary to their cause. Their reasons for galaxy wide genocide go beyond faith. It is a matter of practicality. The Ur-Quan were enslaved once, in the most horrible way imaginable, and will never allow it to happen again. To ensure that no race can threaten them, they must all be destroyed utterly.
The Eternal Doctrine is in oppostion to the Kzer-Za Path of Now and Forever. In the second Doctrinal War between these two schools of thought, had it not been for the intervention of The Captain, the Kohr-Ah likely would have won, thus sentencing the galaxy to extermination.