Here is the full list of all 13 Wild Wild Chess achievements.
Play a chess game
Play 3 chess games in total
Win 5 times against the AI
Play 4 Player vs. Player games
Play 10 chess games in total
Play 15 chess games in total
Win 10 times against the AI
Win 10 times against the AI in Intermediate mode
Finish a match with at least 30 moves
Win 10 times against AI in Expert mode
Win 4 times in a row against the AI
Play 20 chess games in total
Finish a match with at least 100 moves