From studies I have read, I believe it's just their nature but sometimes with a dash nurture thrown in. Serial killers just have it in them. They are screwed up in the head. They enjoy the rush of it all. But the rape side of it could be both nature and nurture. If they were raped as a child then they might be replicating that but if they weren't then once again it's just the thrill of it all.
But that is just my opinion and I'm not a psychologist.

I think that is both with serial killers. They are born pyschopaths or sociopaths and their upbringing determines where they go from there. There are alot of pyscopaths that don't kill for fun and become Drs and ceos, lawyers etc. That is just a general belief and there is always an exception to the rule.

Studies suggest that its both.

There’s also the ‘major head injury’ during childhood element/theory which is very common in serial killers! Plus as someone else mentioned, controlling/overbearing/abusive mother figures?

I think it's a bit of both. I think we all have the capacity for psychopathy (I'm referring to not being able to relate to others feelings/empathy), but people who have had proper nurturing and love are better able to turn away from that.
Think about when you watch the news. Do you well up in tears and break down every time something negative happens in the world? No, because we're able to say "oh that sucks, too bad" and move on. Some people say it's because we're desensitized, but I think it's something different. It's a survival defense.
We're able to turn away because we have that baseline psychopathic response to be selective with our empathy. Otherwise we'd be vulnerable all the time.
I think our parents taught us most of our emotional responses. Not all, but most.
Serial killers, or just killers in general, haven't had that connection established early enough in their formative years, to relate to peoples feelings.

But that's just my guess.

Nurture in most cases. Most serial kills have had pretty funked up lives as children. Its not an excuse for their behaviour though. Lots of people have funked up childhoods and don't go around killing people. Watch mindhunters on netflix