In order to unlock this achievement, no other weapon except for missiles can be used during the battle, regardless of whether or not they do damage. This includes ion weapons and non-damaging beams and bombs. However, damage from asteroids and fire will not invalidate earning the achievement.
In addition, Crystal weapons count as missiles (they are considered projectiles and not lasers), and thus can be used to obtain the achievement.
You can fire a laser weapon but still get the achievement if the defense drone shoots them all down. Use of a Hacking module or a Crew Teleporter is also permitted, and they can be used to destroy the defense drone or enemy systems.
The enemy ship's defense drone does not have to be active when the ship is destroyed to earn the achievement, rather it just has to have one deployed at some point during the fight. This means the enemy's Drone Control system can be taken out to reduce wasted missiles.
This achievement cannot be earned in the final boss fight, because the Rebel Flagship's defense drone is unique and does not function quite the same as other drones script-wise, and because the second phase fight does not end in the ship's destruction.
Typically the tandem strategy is used - fire at least two missiles in near-perfect unison, and the defensive drone will only be able to shoot one down, allowing the other to strike the ship. The Rock Cruiser already comes with two missiles, and enemy ships with defense drones can be encountered as early as the first sector.
In asteroid fields, enemy defensive drones will target incoming asteroids, providing a time window for missile strikes. Targeting drone control or shields (in asteroid field, in first sectors) is recommended. Defense Scrambler augment can be used to great effect, as the augment prevents enemy defense drones of any kind from targeting the missiles you shoot.
Other Rock Cruiser Achievements: