You need to reach a total cumulative score from all your games of 15,000 points. This includes single player, local multiplayer, and online multiplayer games. The fastest way to earn points is by playing local multiplayer against yourself. See "Count of Carcassonne" on how to set this up.
When you play the game, just focus on getting the most points possible for your main player. Use the guest player to help your player get points (build bigger cities, longer roads, complete cloisters). In a typical game, you should be able to earn between 175 and 225 points. At this rate, it will take you around 75 games to reach 15,000 points, taking you at least 15 hours. Along the way you will unlock the "Knight of Carcassonne" achievement for finishing 20 games. Be sure to turn off the River II expansion as it just adds more tiles to the mix, increasing each game's length.
You can check your progress towards this achievement by clicking on "Profiles" from the Main Menu and selecting your gamer profile.