Also, just take short games from a book (Polgar has some in the back), let your child take the winning side. Let him guess some of the moves, and talk about why the players made the moves they did. Keep it simple: threatens the pawn on e5, forks King and Queen, lets the Bishop out, etc.
Unless he's one of the rare prodigies, I wouldn't suggest anything too complicated for him now at age 5. I was only able to show my 5 year old grandson the moves. His attention span was too short for anything else. Just stick with the basic's - value of the pieces, simple tactics (forks, pins, simple checkmates such as the mate-in-one puzzles you've been doing). You don't need the Polgar book yet. You can set up your own mate-in-one puzzles. Chess is first of all a game and should be fun. Let him enjoy it while he's young. Making it like work now could be counter-productive in the long run. Your approach sounds correct. Serious training programs are work, and my opinion is that they can wait till later.
I would press him to find out if he is a prodigy. There is no going back in time, and you've nothing to lose if done responsibly, and he has everything to gain with your assistance. Maybe find out if he is interested in tournament play and interested in the competitive aspect of organized chess.