The PM is appointed by the party, and people vote for a particular party, not a candidate. That was answered earlier. And no there isn't a vice-prime minister.
Well, we have a Deputy Prime Minister which is effectively what the Vice President is really, he takes charge if the Prime Minister is unable to.
John Prescott is our current Deputy Prime Minister.
Listening to him speak is always amusing as it is generally incomprehensible, he sort of goes off on tangents and back in the space of a sentence.
The scary thing is, he was in charge of the country for 20 minutes during Blair's heart scare.