Kologarn's arms have different attacks and can be targeted as separate mobs, and once an arm is destroyed damage equal to the arm's maximum health is passed on to Kologarn's main body. Each is periodically destroyed during the encounter, but regenerates after sixty seconds.
To get this achievement, both arms must be destroyed within a short time and Kologarn himself killed. It is not as difficult as it may seem, as destroying both his arms when he is at low health can easily pass on enough damage to kill him when the second arm is destroyed.
Focus DPS on his more dangerous right arm (on your left) to destroy it as quickly and as often as possible, and have DPS switch to his left arm (on your right) whenever his right arm is down after killing the spawned elementals. With some luck you'll near the end of the encounter with both arms at low health, which is the key to getting this achievement.