Snopes has a good page on this:
Chaplin did indeed fare poorly in a Chaplin look-alike contest, but the competition took place in a San Francisco theater. His final standing is not recorded, although it was noted that he "failed even to make the finals." Chaplin told a reporter at this time that he was "tempted to give lessons in the Chaplin walk, out of pity as well as in the desire to see the thing done correctly."
The sources listed on the Snopes page are:
Chicago Herald. 15 July 1915.
The relevant quote from the book is:
Vaudeville houses still showed motion pictures in 1915, and many were promoting the Chaplin vogue by sponsoring amateur Charlie look-alike contests. Among the early winners was Bob Hope, who took first prize in a Chaplin contest in Cleveland. Charlie himself was not so lucky. When he entered a contest run by a theater in San Fransisco, he failed even to make the finals. "I am tempted to give lessons in the Chaplin walk" he told a reporter, "out of pity as well as in the desire to see the thing done correctly".
The only citation listed for this passage in the book is the very same Chicago Herald for July 15, 1915, Vol. 11, RLC.
I am unable to verify the second reference at present, as I am unable to find any archive online for the Chicago Herald, which seemed to only exist under that name for four years. Short of going to a library with access to the archive there does not seem to be a way to verify this reference.
Edit: This answerThis answer records that the article could not be found.