Click on a chess piece. A green dot shows up, and you click it as the place for your piece to go. I haven't coded a check system, so just watch out for them. Checkmate is done when you remove the king, not when the king is unable to get out of check.
I coded chess as a little challenge for myself. Report any bugs in the comments. I will make a sequel that has an AI, but that will come out very later in the future.