According to Mr. Darcy's own words to Elizabeth, his greatest fault is that he claims to suffer from a "resentful temper". In his own words, he said:
My temper would perhaps be called resentful.
Darcy spoke these words during a conversation with Elizabeth. As we know, mostly every dialogue between Darcy and Elizabeth ended up quite awkwardly. Elizabeth's extroverted and outspoken personality contrasted dramatically to Darcy's introverted and quiet demeanor.
During this conversation Elizabeth attempted to tease Darcy about his behavior. The main point that she was trying to make is that nobody should be so proud as to pretend to have no flaws. To this, Darcy responded with the quote mentioned above. Perhaps Darcy was also sending a warning to Elizabeth that said subtlety: "Quit bothering me, for I will not stop being angry at you". Perhaps he felt a bit of shame in admitting that he holds grudges against his enemies. Either way, Elizabeth was surprised to see that he admitted to that much.
Mr. Darcy has been considered by Elizabeth as one who has musjudged the love between Jane and Mr. Bingley, for example. However, Mr. Darcy belives that his only fault is not being able to talk to a person, when he doesn't know him/her. He even tells this to Elizabeth while she is at Lady Catherine's house.
The readers, however, have a different view about this at the beginning of the story, where pride seems to be his gretaest fault. But Mr. Darcy belives that what he has done has been out of good will. For example, when he apparently saves Mr.Bingley from a bad marriage. Only later in the novel do we really find out Mr.Darcy's character and how he has been, infact, been 'generous' to Mr. Wickham and how he meant no ill-will towards Jane. Elizabeth does not approve as her sisters love is taken away. But Mr.Darcy does bring Mr. Bingley to Longbourn by the end of the story.
Mr. Darcy feels his fault was when his sister was about to elope with Lydia. Moreover, he also feels it was his fault for not talking to Elizabeth and explaining to her what really happened between him and Mr.Wickham. He also believes that it is his fault, when Lydia elopes with Mr. Wickham.
In general, there are many points in the novel where one anlyses and agrees or disagrees with the characters. In Mr. Darcy's case also, at some point the reader agrees or disagrees with him. There are also parts in the novel where one character brings out the character of another. Hence, Elizabeth is a device of characterisation for Mr. Darcy. She tries to read Mr. Darcy's character. Also, at the beginning of the novel, it seems Elizabeth and Darcy are foils to one another but later on in the novel its quite the opposite.
When Elizabeth determines Mr. Darcy's fault or her own, it also in a way shows how Mr. Darcy approaches his own character and Elizabeth's. It is very interesting for the readers.
NOTE: I have kind of given you the main points. I have written and given examples from the book and you can support it from the text. Give quotes to show his character and tell how he has determined his character and explain it further.