To earn any of the achievements from this title update, after loading my save file, I went to the arena in the top right, selected the Tournament of Puzzles pack, then started playing matches.
In order to fill up an entire Tic-Tac-Toe board, you'll need help from your opponent, that can be another player or your second controller. I did this with a second controller. Have one player place four tiles in a way that they don't have three in a row, and have the other player place five tiles, avoiding three in a row until they place that fifth tile. I placed four tiles in a 2x2 square, then had the other controller fill in the other spots, saving the corner for last. When I did it, I won the match, I don't know if it matter if the board ends in a draw.
For this game mode, as you complete puzzles you'll usually l get a board piece for the tic-tac-toe board, sometimes you'll get a key instead and sometimes both, it depends on the level. Once you get your piece, go place it on the board, you can only carry one piece at a time, though you can solve all the puzzles and just leave the pieces sitting there.