The achievement is unlocked when you kill the 500th enemy with an ability rather than a weapon. Before earning this achievement you would already have unlocked:
The only difference between this achievement and
Shock and Awe is the number of enemies killed. You are required to kill 100 enemies in the earlier achievement and 500 enemies for
And Then There Were None.
Each class has abilities that can be used for kills that count towards the achievement. Some examples are Sorcery's storm bolt & ice barrage, Might's quake, and Finesse's Shadow Flare. The ability is mapped to a button for quick access during combat. It is activated by pressing

and the button to which the ability is mapped.
Abilities lack power at low levels but become quite lethal when enhanced. Nevertheless, the achievement can be unlocked using low power abilities by whittling down enemy HP with weapons and administering the coup de grace with the

+ability button attack. Even though the majority of the damage was done with the weapon, if the enemy succumbs as a result of a final ability strike, the kill will count towards the 100 tally required by the achievement.
Alternating between weapon attacks and ability strikes is a good technique. The reckoning gauge recharges faster when switching between weapons and abilities. Abilities tend to interrupt enemy attacks and/or slow them down. If one gets in the habit of fighting in such a manner, the achievement will be unlocked through normal game play and there will be no need to grind out the achievement.
In the below video, the player character is at level 28--a hybrid sorcery/finesse destiny of Spellcloak. The destiny provides +30% elemental damage. Storm bolt and ice barrage enhanced by chain lightning and frostshackle respectively are even more lethal with the boost in elemental damage. They provide effective crowd control interrupting enemy attacks. The player character, though vastly outnumbered, sustains minimal damage. This is a very tedious achievement if you grind it out. I recommend unlocking the achievement through normal game play by incorporating abilities into your tactics and becoming more survivable and lethal in the process:
Note the achievement tracker in the video's lower right hand corner. You can use it to check your progress towards the achievement.