Here is the full list of all 38 Chess Ultra achievements.
Beat an opponent with an Elo rating higher than yours
Using a Knight simultaneously attack the enemy King and Queen
Castle queenside
Pin a piece against your opponent's King
Win a game
Advance a pawn to the eighth rank
Sacrifice the Queen then go on to win the game
Place a White Knight on e6
Draw a game by stalemate
Have two pawns of the same colour on the same file
Make a capture or exchange
Castle kingside
Put an opponent in check
Develop a White King Bishop to b5
Stack two Rooks one behind another in the same file with the Queen at the rear
Deliver checkmate by a Rook or Queen along a back rank, with the King's escape blocked by friendly pieces
Promote a pawn to a Rook, Bishop, or Knight instead of a Queen
Perform an en passant capture
Deliver a check by two pieces at the same time
Complete all the tutorials
Capture 100 pieces
Deliver checkmate where the King is blocked on both sides by his own Rooks
Win a game in three moves
Win a game in four moves
Win a tournament
Complete 50% of the chess challenges
Create a random online game
Issue an online challenge to a friend
Play a game using the Fischer timer
Beat an opponent with an Elo rating more than 75 higher than yours
Win 5 tournaments
Manoeuvre a pair of Rooks onto your opponent’s second rank
Deliver checkmate by two crisscrossing Bishops, with the King's escape blocked by friendly pieces
Win a 32 player tournament
Beat an opponent with an Elo rating more than 150 higher than yours
Beat the computer on Grandmaster difficulty
Complete all the chess challenges
Win 20 tournaments