Your story must be perfectly readable and understandable by people who do not play chess, do not know the rules, and only know through pop-culture osmosis that there are pieces called 'rook', 'knight', etc. Write with that in mind.
With that in mind, I probably wouldn't use chess notation at all. Somebody who has never played chess wouldn't be able to read it. Instead, I'd describe the situation, in broad strokes. A player might be forced to sacrifice a piece, or they might suddenly realise their careful plan has a fatal flaw, the opponent might find a way to escape a trap laid for them, or they might be playing an aggressive game, forcing the MC to do nothing but react. Those are all evocative descriptions that do not require the specifics of what's happening on the board, to be understood.
@Stackzebra mentions in a comment the chess game from Harry Potter and the Philosopher's Stone. It's a good example. The scene is engaging, it's fun, the reader experiences the tension of it, it works. Only trouble is, if you put the titbits of information about the game together, it turns out that not only is there no actual position in the author's mind that she's describing on the page, but she appears not to know the chess rules at all.
If I take one step forward
says Ron, the knight.
This example is great because the chess fails: the scene works despite the chess failure. Exemplifying what is important (the character dynamics, the sense of danger, etc.) and what isn't important (the chess).
For a different example, I would point you to the videogame Dragon Age Inquisition. While your player-character is walking around doing things, some of your companions banter in the background. And two of them start a chess game. Here's a link to the full dialogue, with added animation, and added overlay of the actual game being played. Incidentally, here the chess does work - they are playing the Immortal Game.
An example of dialogue from their game:
Solas: So, where were we? Ah, yes. Mage to C4.
Iron Bull: Little aggressive. Arishok to H4. Check.
Solas: Speaking of aggressive. I assume Arishok is your term for the Queen?
The two characters come from different cultures, their names for the pieces are different. It's a titbit of worldbuilding information that's interesting whether you're following the game or not. Also, note the commentary about a move being aggressive. Again, that clues in the non-chess-player audience. The game proceeds in the same way: it's all audio that you hear while playing, no board before you, so the dialogue must engage you in other ways, similar to how a novel would have to do it. In order to achieve that, every move is commented on, in a way that one doesn't need to understand chess in order to understand what's going on.
Iron Bull: You've got no Towers. You're down to a single Mage. Too bad you wasted time moving that Pawn to... to... You sneaky son of a bitch.