Rupert Murdoch would be happy with Hillary as president. He gave her money during the 2008 primaries. It's part of his plan not to put all his eggs in one party basket (hence his support for Tony Blair in Britain).
That's not to say that Fox News is liberal or that it's secretly working against conservatism, just that they're far from the big bad conservative machine liberals have nightmares about and/or want people to think is so awful.
I think it’s pretty clear Fox News is anti-conservative.