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.