In chapter 10 we find the Bingley party assembled at Bingley's drawing room. This occurs during the period of time when Jane becomes ill during a visit to the Bingley's and is told to remain with them to recuperate. Elizabeth is there to take care of her sister. Darcy is there too, since he had been accompanying the Bingley party for sometime now.
The conversation which prompts the quote by Elizabeth:
You have only proved by this that Mr. Bingley did not do justice to his own disposition.
occurs during a disagreement between Bingley and Darcy over the "appearance" of humility in people. Darcy insists that Bingley has a tendency to downplay himself, not because he is humble, but because he enjoys to flaunt his strengths as well as his weaknesses. Darcy does not say this to hurt Bingley's feelings more than he does to expose one of Bingley's hidden traits- an insight which the reader appreciates considering the flat nature of Bingley's character.
However, there is more to this disagreement. Darcy is actually mad at Bingley for not having left the countryside as quick as Darcy would have wanted to. Hence, Darcy was being his usual self: difficult and sarcastic.
When Elizabeth responds with the quote in question, she basically tells Darcy that, by exposing this trait about Bingley, he is confirming that Bingley is actually a fellow who just enjoys to humble himself, not trying to play mind games like Darcy intends to say.
Perhaps Elizabeth, who is still a bit prejudiced against Darcy, considers that, if Bingley were a show off, he would be flaunting excessively or being a snob to everybody else the way that Darcy would.
However, Charles (Bingley) corrects Elizabeth by saying:
I am exceedingly gratified," said Bingley, "by your converting what my friend says into a compliment on the sweetness of my temper. But I am afraid you are giving it a turn which that gentleman did by no means intend;
We know that there is tension in the air. Bingley is quite happy to stay in the country. After all, he likes Jane and wants to look after her. Darcy, on the other hand, wants Bingley away from Jane and to return to London as soon as possible. Bingley is not blind to this fact but, by his reaction, we can tell that he has been trying to deflect Darcy rather than to confront him.
However, it seems that Bingley has had enough of Darcy at one point, for he says
I declare I do not know a more awful object than Darcy, on particular occasions, and in particular places; at his own house especially, and of a Sunday evening when he has nothing to do.
We know that this really makes Darcy mad, and that he hides his feelings with a smile. Elizabeth notices that and tries to downplay the situation. However, this goes to show that the character of Darcy is meant to be complicated and annoying. At least Bingley sticks to his gun during this argument...for now.