The first thing I suggest you do is ignore this Achievement. The counter for this Achievement runs behind everything you do, so if you ignore it at first, you make progress without trying. In this way, you get this Achievement faster.
Start off by getting the Achievements related to points in the three modes (Marathon, Timed, & Survival). Unless you are crazy good at this game, or very lucky, you won't get these Achievements on your first try. As you attempt them more, your total game count goes up, getting you closer to Hexic Addict. At the absolute least, you'll have 3 games already completed, taking off a small percentage of time you'll need to play to boost.
You can also play Marathon games while trying for the Big Cheese of the South Seas and Oyster-Meister Achievements. These are extremely hard to get (unless you're really good, which I assume you aren't for the purposes of this solution) and so each attempt will get you closer to both goals.
If all you want to do is get this Achievement as fast as possible, you should do what others have already suggested, which is start a Timed game and make one point-scoring move. This will start the 50 second timer, and then the game will end, giving you one more completed game of Hexic. You'll have to do this up to a maximum of 100 times, and factoring in the time it takes to set up each new game and make the first move, it will take roughly 100 minutes, or 1 hour and 40 minutes.