What symbols would each character symbolize in a game of chess?
An "Endgame" in chess is when only two kings are left on the board. Neither can win or lose. All that happen is that they can endlessly move around, and around, and around, never putting each other into check, and the game only ending when it is aborted as a stalemate.
In that sense, Hamm and Clov are both kings, unable to finish anything (note the first and last lines of the play!) moving around the chess board.
But, it's important to say that the characters share characteristics which are perhaps supposed to suggest other chess pieces. Hamm (who sometimes openly utters "Me to play") is most like the King, a very weak and vulnerable, but also a very powerful piece: to defeat the king is to win the game. Like a chess king, his movement is restricted, and he relies on another piece to protect him (Clov!)
Clov's odd movement patterns and his staggering walk seem closest to the odd "L" movement-turns of a Knight. Nagg and Nell are weak, and unimportant pawns, the loss of whom is insignificant. Nell's death bothers no one.
But the key chess point is that, ironically, the "Endgame" means the game can never end, but must continue and continue and continue.
'Endgame' in chess is hard to pinpoint. It is the final stage, after most of the bishops, knights and rooks have been lost in tit-for-tat clashes. Endgames between skillful players (which Beckett was) often consists of only pawns and kings.
Hamm is the white king, trapped in a defensive position behind stalemated pawns. He is unable to move, pinned in by the still mobile, black king, Clov.
Hamm can only delay and hope for stalemate. Clov senses the chance of victory but is frustrated.
The link below explains all the chess images/references in the script. It is a very very long article and the first part is about very complicated chess. Don't worry about the chess section, scroll all the way down to the bottom and you'll find lots of stuff from the play and how they relate to a chess game between Hamm and Clov.