I have managed to get him to kill some of his minions by standing in the mob and not doing too much damage, but it takes way too long just for a couple of kills. What is the ability he uses that can kill his own minions (what does it look like), and how do you make use of it to get the achievement?
The skeleton king has a couple of attacks you should watch for:
weapon swings - these are pretty generic, he just swings his mace around. These strikes can damage his minions if they are too close to him.
Whirlwind - this is what you want to watch for if you're doing the achievement, however, as it deals a good deal bit more damage (both to you and the minions).
The achievement can be earned pretty easily by kiting the skeleton king around, making sure that there are always minions between him and you. Equip gear with Life Regen or Life Steal and simply run around the arena, regenning life between attacks.
Try to avoid damaging the skeleton king's minions yourself (or at the very least, avoid killing them), because the more of them there are, the more likely it is that the Skeleton King will hit at least one.
