I've managed to get my quest added into the game, and I am able to start it. But I can't for the life of me work out how to make the quest advance after killing the target NPC. Would anyone be able to shed some light on the matter?

Add a script to the NPC in question:

Begin OnDeath  ;so it will run once when the NPC diesIf GetStage YourQuestIDGoesHere == XX  ;replace XX with number of the stage you should already be onSetStage YourQuestIDGoesHere XX  ;replace XX with the stage you want it to be nowEndIfEnd

You may need to add an "Else" with stuff to do if the NPC dies before your quest is at the right stage, or set your NPC to essential until the right stage.

I believe that the stage in your quest could alternatively have a condition which must be met for the stage to begin.

GetDeadCount [Parameter] == 1

Change the parameter block to the NPC.

So, for example, if the NPC was AnvilGuardPatrolDay01, then it would be like this:
http://yourimg.in/j/ajbnh1.png