After you're past level 10, you need to level up two games in a row, but you don't have to level up in the same mode.
I recommend playing one mode until you're certain that you can level up after the next game.
Then go play another mode until you level up.
Finally, switch back to the first mode and level up to unlock your achievement.
It shouldn't matter which mode you do, but I leveled up in Klondike the first game and Spider in the next.