The fastest way to do this is to set up a single player game on easy mode with AI turned off. Then just keep getting the answers correct! Because you are playing single player, you can pause the game without penalty, which will allow you to stop the time limit and Google any question you don't know. (In easy mode, it's multiple choice, which simplifies things. In other modes, you have to manually type it.)
The key to getting this quickly is to accumulate as much money as you can when you get Daily Doubles, then bet it all and get it correct to double your money. When choosing questions, I suggest you start with all the $1,000 questions in the first round (and $2,000 questions in the second round) then work your way up, that way you will have the highest amount of money possible when you hit the DDs.
Then once you get to Final Jeopardy, wager it all and make sure you get the answer right (again, by pausing after the question is read and looking up the answer online). If you follow this method, you'll have the achievement in just a few games!