To dissemble is to speak without telling the truth, which sometimes entails lying and sometimes instead entails a creative set of strategies aimed at simply avoiding the honest truth.
The narrator is practiced in acting and/or evading the truth.
"He is a scholarship student, but he longs to appear to be from the class of wealth and heritage. He bends all of his wit to create the impression that he is. He is, in other words, an strikingly adolescent mixture of ambition and pretense" (eNotes).
When Wolff's narrator admits to some dishonesty on his own part but is surprised to find that Bill also has experience "dissembling" too, he is simply saying that he did not expect Bill to be the kind of person either willing to lie or, perhaps, needing to evade the truth.
In commenting on the honesty of others and his own honesty, the narrator helps to establish one of the novel's central themes -- the performance of identity.
Many of the writers presented in the novel (from Robert Frost to Ayn Rand) serve to articulate this theme as they appear to be both true to their reputations and something else besides. The conceptual ambivalence of identity connected to these figures informs and comments on the narrator's own shifting sense of self as well as that of the school's headmaster.