Is this quest dependant on my Destruction skill? Neloth won't give it to me.

It shouldn't be.

You do have to do the experimental subject quest first though, the one where he asks to cast a spell on you, then you go out and explore, and then report back to him when you notice any side effects.

Dang. I did that quest a while back.

Well he just gave me the quest after I levelled my destruction to 47. Not sure if that's what did it or what.

I think Neloth's quests may be radiant. I don't think there is a set order in which you get them. Actually, I was surprised at how many different quests he was giving me. I think I was a test subject at least twice. Also a couple of fetch quests. Black book quests. A kill quest. Lots of quests.