Moved sign to the begining of the method (sign will always be 'K', ideally should be constant)
Added string command variable to hold the full directional order. (should ideally be passed as a string in the first place).
Removed most of the body of the conditional direction check, the only thing that remained was the update of the newCoords position and the assignment to kMoves.
Conditional check was changed to a switch.
The body was move as a seperate if/else construct as it was always the same execution.
Proverka call was moved from the conditional statement check to a boolean.
Changed the method name to CheckNextKingPosition