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Both Mr. Darcy and Mr. Bingley are close to each other even though they see the world from different views. Both Darcy and Bingley fall for the Bennet sisters, Elizabeth and Jane. Mr. Bingley has a manner like Jane Bennet's, they are fond of everyone, see only the good in people, and are very pleasant characters. Mr. Bingley is quick to approve of everyone and everything he sees. He loves to dance and socialize with several people, and attends many balls. He's affable and comfortable around everyone. Mr. Darcy, initially, is almost the opposite. Mr. Darcy, however, becomes more like Mr. Bingley as Elizabeth is able to break down his exterior of pride and social status. Mr. Darcy becomes quite fond of Mr. Bingley, he sees in him what Elizabeth is talking about.
We are not told HOW Mr Darcy and Mr Bingley met to become friends - we can only guess that perhaps they were at school at the same time or something like that. (Although even if that is a reasonable explanation, Bingley may have been Darcy's fag or gotten tutoring from him -- some "unequal" relationship; because Darcy is 5 years older than Charles Bingley.) At any rate, when the book opens, they are already "great friends" (which is also before they even meet the Bennet sisters) but we are never told in what manner the first grew into friends.
We are told by Miss Austen that the REASONS they are friends now include that Darcy appreciates Bingley's open, easy and amiable manner to everyone; while Bingley has great respect and appreciation for Darcy's intelligence and judgment, Darcy acts as an advisor to Bingley in many areas of his life. He improves Bingley's mind, while Bingley adds a little lightness to Darcy's life.