Fox News is owned by a billionaire and it operates in his interest(s). I don't think that Fox, as an entity, hates the President. Rather, it is putting forward an agenda that benefits a rich and powerful man and the rhetoric it uses to characterize the President is simply propaganda. It's just business. It's like the hitman who didn't hate his victim, but killed him because the killing was in the interest of the business the hitman worked for.