How does Oscar Wilde present the characters of Jack and Algernon in The Importance of Being Earnest?

Wilde presents these characters via indirect characterization.  No one directly describes Jack or Algernon; readers (or viewers) must mine their conversations for clues about their characters and motivations.  It is not difficult, given the nature of their first conversation with one another in the play.  For example, Jack insists that, "When one is in town one amuses oneself.  When one is in the country one amuses other people.  It is excessively boring."  Then, a few lines later he declares that all his neighbors are "Perfectly horrid!" and that he never speaks to any of them.  Contradictory statements like these help us to understand that Jack is a somewhat ridiculous character.  He doesn't seem to really mean anything he says because he is just as liable to contradict it in the next breath.  Likewise, he admits (after the debacle with the cigarette case) that his name is actually Jack when Algernon has only ever known him as "Ernest," because he has invented a younger brother by this name so that he has an excuse to come to town and do all the things Jack cannot do (i.e. gamble, drink, and so forth).  Jack feels obligated to maintain a high moral tone because he is the guardian of a young woman named Cecily, but -- via Ernest -- he can get up to all the somewhat less moral behaviors in which he longs to indulge in the city.  Thus, we learn that he is quite dishonest and has a bit of a dark side that he hopes to conceal forever from his family and friends in the country.

We learn about Algernon's character in a similar manner.  We know that there are cucumber sandwiches at the beginning of Act 1, and he will not allow Jack/Ernest to eat them because they have been specially prepared for Algernon's Aunt Augusta.  However, Algernon eats them all himself throughout the course of the scene!  Then, when Aunt August arrives for tea, Algernon seems seriously displeased and surprised that there are no cucumber sandwiches when he has actually eaten them all himself.  We learn, then, that he is likewise ridiculous and dishonest.  He has also invented an friend, named Mr. Bunbury, so that he can escape Aunt August as often as he wishes, saying that he must attend his poor invalid friend.  He and Jack are very similar in these ways.

It is through listening in to their conversations and the utterly ridiculous statements they make about social convention, love, marriage, and the like that reveal to us who they are.