MOST IMPORTANT: I played the game a lot with my girlfriend, and it seems that locally, the score only counts for the total of the player that selects multiplayer in the menu, sets the match settings and starts the match. Other local players will NOT have their score added to their total. In addition, I suspect that your total score is saved separately on each console you play and NOT added up, so playing on multiple consoles may delay getting the achievement.
A little addition to the method mentioned above by Vermin360: If you have the King and Baron DLC, play king and baron matches against one easy CPU or a second controller. King and baron titles add a number of interesting tiles, as well as scoring points for completed cities and roads at the end of the game, greatly increasing scores for the round.
Also, avoid using the River/River II DLC. The river tiles will add little score to you in the end, and mostly the river will make cities and roads harder to complete. Farmland is also more divided so you will need to farm more for a comparable amount of score, and matches will take longer.
There is no way to keep track of your running total of points, other than manually noting it. Given an average of around 160-170 points per match, it will take some 30 matches for this achievement to unlock. This average is very easily achieved if you follow the above advice.
Now that you know all problems and solutions concerning this achievement, have fun getting it!