Chess Ultra

The quickest way is to either dual-box, use two accounts, or have a friend to help you.

It doesn't matter if you play as black or white.

Basically, have your queen take everything of the opponent except for the King. Have them move the King to a corner (A1, H1, A8, or H8). With the King in the corner move your Queen to one of the following squares:
King in A1 = Queen in C2 or B3.
King in A8 = Queen in B6 or C7
King in H1 = Queen in F2 or G3
King in H8 = Queen in F7 or G6.

Putting the queen in this position will result in the King not being in Check, but being unable to move to a square on the board which does not put him into Check, resulting in the Stalemate.

Be careful not to checkmate the king when moving your queen into position or taking pieces!

Stalemate

Stalemate is when the enemy cannot make a single move but the king remains alive. This can actually be obtained with any number of pieces, stalemate does not mean no one can win, it means one player cannot make a move.

What I did to get this achievement is fight a Novice AI, took out all the pieces other than the king and pawns. Make sure to save both your rooks because they are key to obtaining stalemate easiest, you can just use 1 rook but its more difficult. Once I had control of the board with the AI only being able to move with the king I pushed the King into the corner without obtaining checkmate (where the king is in check and unable to move). Instead you want to merely not allow the King to move, at all. So the idea is to cross your rooks attack paths placing the king in the corner, which will create stalemate.