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