In Charles, how do we know who Charles is?
We do not know for sure who Charles is -- we are not explicitly told. But we can easily infer that Charles is, in fact, Laurie himself.
The best way to tell this is from the end of the story. In the time leading up to the PTA meeting, the narrator and her husband are very interested to meet Charles' mother. They want to see what kind of a woman could produce and raise a child like that.
When Laurie's mother finally talks to the teacher, we can see that the teacher reacts to her in the same way that Laurie's mother might have reacted to Charles' mother. When the teacher says that they are all "so interested in Laurie" it shows that everyone wonders why Laurie acts the way he does.
This is the clearest indication that Charles is Laurie -- the fact that the teacher talks about Laurie in that way.