Leela challenges you to play games against her three times: Once with Air Hockey, once with Scram, and once where you get to choose between the two.
While I found Scram to be the easiest of the two, I believe this achievement only counts the Air Hockey matches: AFAIK it's the only game of the two that has any change to it depending on how much you play it, as it gives you mallets in new colours the more you play: First you have access to red and blue, then you get purple, and then you get green. The green mallet is also what is depicted in the achievement image.
So just play Air Hockey against Leela until you get the green mallet(should be around 6-7 rounds, though you might have to win that many), and then play/win a round with the green mallet for good measure.
Then just play through the game, and the achievement should unlock at the end!