Which approach for user classification on chat text (classifier, representation, features)?
You're asking what ML representation you should use for user-classification of chat text. User-classification is not the usual text-processing task.

It's not strictly necessary to semantically understand what the user is saying, only how they're saying it; so we look for telltale features indicative of a specific user. And we don't necessarily need to use, or solely rely on, the usual text-processing representations like bag-of-words, word-counts, TFIDF and word-vector.

Here are some features which are predictive of the user:


character length, word length, sentence length of each comment
typing speed (esp. if you have timestamps in seconds)
ratio of punctuation (e.g. 17 punctuation symbols in 80 chars = 17/80)
ratio of capitalization
ratio of numerals
ratio of whitespace
character n-grams (and notice these can pick up e.g. l0ser, f##k, :-) )
use of Unicode (emojis, symbols e.g. stars)
ratio of specific punctuation (e.g. how many '.', '!', '?', '*', '#' )
word-counts, esp. anything statistically anomalous, foreign, slang, insults
anything else you can think of that seems predictive for these two users, e.g. number of misspelled words per sentence (may be actual typos, or come from predictive swiping on a cellphone)