There’s probably a lot of different ways that you could do something like that, but it sounds like from the line of questioning, you need a handle on how Game Mode works, how Pawn works, how Actor works, maybe how Movement works … There’s a whole lot of questions here that need to be a little more specific for anyone to really be able to provide an answer.
Game logic generally goes in Game Mode, and what I’d do is I’d write the entire chess game logic (rules of chess) into the Game Mode, all the board state handling into Game State, controls in PlayerController, and movement probably somewhere in a Pawn (unreal pawn not a chess pawn). I’ve done this in UE3, and it works fine, although I suspect there’s a lot more to UE4 that could be learned to do the same thing.