I will have him killed.
Would mean that I will arrange for someone to be killed.
He will get killed.
Would mean that some circumstances would lead to someone's death.
Now, in terms of how we would talk about the real world, "I will have him get killed" would be strange and unwieldly - just about justifiable, but pretty bad phrasing.
In talking about producers of fiction (specifically, the writers of Dexter in this case), there is a remove between the effect of characters upon characters - such as say a policeman character who kills Dexter - and the writers who have control over the circumstances Dexter encounters in a way that no one does over anyone in reality. As such:
I will have him get killed.
Makes perfect sense in reference to a fiction, and means I will write the story in such a way that he gets killed.
The meaning of the quoted question
Why can't they have him get caught or killed?
Follows from that, why can't the people who produce the show write the story in such a way that he gets caught or killed?