How to insert values in mysql taking previous row into account
In any case you'll need some sort of sequence. As MySQL does not provide such things, as far as i know, the only resort is an AUTO_INCREMENT column. Put some in a table abused as a sequence. Than have a BEFORE INSERT FOR EACH COLUMN trigger on your game table. Inside it, insert into the sequence table, so that a new incremented value is created. Select the maximum value of the sequence table. You can now calculate your values for player end round:

Let v be the selected value from the sequence table. Then the player p = (v - 1) mod 2 + 1 and the round r = ceil(v / 2).

But be aware, that deleting rows from your games table will not reset the sequence table. The pattern will continue on where it left the last time an insert into the games table was made.

Example:

CREATE TABLE game (id INTEGER AUTO_INCREMENT,
                   player INTEGER,
                   round INTEGER,
                   PRIMARY KEY (id));

CREATE TABLE game_sequence
             (value INTEGER AUTO_INCREMENT,
              PRIMARY KEY (value));

DELIMITER $$
CREATE TRIGGER game_ai
               BEFORE INSERT
               ON game
               FOR EACH ROW
BEGIN
  DECLARE _value
          INTEGER;

  INSERT INTO game_sequence
              VALUES ();

  SELECT max(value) INTO _value
         FROM game_sequence;
  SET NEW.player = (_value - 1) % 2 + 1;
  SET NEW.round = ceil(_value / 2);
END;
$$

DELIMITER ;