There is no hiding it. Playing a total of 1000 games on Connect 4 is a horrible, mind-numbing achievement. Fortunately, a way has been discovered that allows 1000 games to be completed in a slightly bearable time frame.
The fastest way to earn wins or complete games is to play the challenge that is located in the second column from the left and one down from the top called "Puzzle." This challenge only requires that you place a single chip to win. You MUST WIN the challenge for this method to work. Placing the chip in the incorrect spot will not work. Just keep loading up the challenge over and over, repeatedly placing the chip in the winning spot (there are only 3 variations of the puzzle, so this is easy). This method should in theory go very quickly, but unfortunately the game's menus are very laggy and doing this method the game tends to bog down and often crash. Still, with the lag and crashing, you can complete between 200-250 games per hour. Meaning to grind 1000 games it will take you around 4 hours after you have completed the miscellaneous achievements. It is also important to mention that the achievements will NOT pop while winning challenge games. After surpassing the requirements for the achievements, you MUST play (and possibly win) a quick game mode match for the achievements to unlock.
You can check your progress towards these achievements from the Play Menu by clicking on "Your Stats."