I'm going to ask my friend not to talk about a computer game
Why don't you want him to talk about the game?
He just goes on and on about it all the time.
There is nothing wrong about your friend talking about the game. He is allowed to talk about it if he wants to. If you don't want him to criticize you, that is different.
I'm going to follow him around and see when he's talking about me.
It is unrealistic to follow him around and see when he is talking about you. There is no reason for you to do that, it should not bother you so much.