First, explain a little about your game. Let's take Super Mario Bros. as an example.
"Everybody knows Super Mario Bros, right? It's a game with two dudes that rescue a princess that keeps getting kidnapped by an evil turtle."
Now, find a reason for your character to get the game. It can be anything from nostalgia, liking the series and wanting to play the original to see how far it's come, or just really wanting to play video games.
But before we get to the game, the character needs to get it from some shady place, ranging from a website nobody's heard of, to some store that just happens to be closing.
Now you're ready to get your character playing the game!
Make the character play like normal, and then something really spooky happens, like blood in games without blood, or worse, knowing the player's name!
Then pass all that off as a glitch.
Keep adding to the spookiness until the game character gets killed and can't come back, no matter how many times you reset the game.
Your character needs to be shocked by this or your pasta sucks, it's a rule.
Have them give up and actually be affected by the game.
"I'm never playing another Super Mario Bros. game again."
Then make them die or be haunted by a ghost that was in the game all along.
Then the ghost kills them.
But the story's still on the internet because the ghost is very kind and won't just leave all that work unposted.
Good luck writing!