I want to give the player an achievement when they kill a specific mob, but I don't know how to do things when a mob dies.

Events are your friend.

Events are your friend.

 

I've tried LivingDeathEvent

@SubscribeEvent
public void onEntityDeath(LivingDeathEvent event) {
	System.out.println("Entity Death");
	Entity e = event.entityLiving;
	DamageSource source = event.source;

	if(source.getSourceOfDamage() instanceof EntityPlayer) {
		System.out.println("Player Kill!");
		if(e instanceof EntityZombie ) {
			System.out.println("Zombie Killed!");
			mobZombieKill = true;
		}
	}

}

@SubscribeEvent
public void onPlayerTick(TickEvent.PlayerTickEvent event) {
	if (mobZombieKill == true) {
		System.out.println("Achievement");
		event.player.addStat(ExtraAchievements.mobKillZombie, 1);
		mobZombieKill = false;
	}
}

and none of the "System.out.println"s get triggered.

Have you registered the event?

MinecraftForge.EVENT_BUS.register(new YourClass());

Have you registered the event?

MinecraftForge.EVENT_BUS.register(new YourClass());

I use

FMLCommonHandler.instance().bus().register(new AchievementEvent());

Unsure if it makes a diffrence

Big difference. If the event is in cpw's packages, use what you are using. If it is from Minecraft forge packages, use what I said.

As mentioned there are different event buses.  In my tutorial on events I explain which events are on which busses, so it might be interesting to you (either this time or in the future next time you are considering using events): http://jabelarminecraft.blogspot.com/p/minecraft-forge-172-event-handling.html