Easiest way is a taxi, of course. Total travel time would be 10-12 minutes. I don't know about the fare, as I always rent a car at WDW, but I don't think it would be more than $15.
But the cheapest way (in fact, free) is not much harder - take a bus from Beach Club to MK, then take the Resorts monorail line around to Contemporary. Total travel time would be about 40 minutes, not counting the initial wait for the bus to pick you up at Beach Club.
None of the monorail lines actually stop inside the parks. The MK monorail stops are all outside the gates, and the monorail doesn't even go into the park. The Epcot monorail actually loops through Epcot, but its stop is also outside the park gates.