Exactly two years, by the end of the 5th season.
Season 1 starts off with Walt's 50th birthday.
When Walt comes back to Albuquerque in the end of Season 5, he starts of the day by celebrating his 52nd birthday:
The show ends on the same in-universe day.
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In the pilot, we see Walt celebrate his 50th birthday
(this is where we see the "bacon spelling out his age" thing for the first time).
In S5E4 Fifty-One , we see him celebrate his 51st birthday.
In S5E1, we see a flashforward where Walt is celebrating his 52nd birthday alone in a diner.
In the last episode, we see the flashforward resume, so the show ends around Walt's 52nd birthday.
Beyond that, I'm not sure how long each season was, but those are some anchor points for how much time has passed during the show.
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Perhaps it's because I'm a mother of very small children, but it helped me to look at the age of Walt's new daughter to help me somewhat decipher the passage of time in the series.
Skyler was pregnant in the first episode, and the daughter was born in the second season.
She's still an infant, which means to me that a year hasn't passed yet since her birth.
Watching the little girl grow might help to determine how much time passes in each season.
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In S02E13 , Jane's father reports that Jane was born on April 4, 1982.
In S03E01 , the TV news reports that Jane was 26 years old.
Assuming these are are correct and we are using the standard US convention of reporting ages, then Jane's death took place some time between April 4th, 2008 and April 3rd, 2009.
In S05E08 , one character said, "I mean, whacking bin Laden wasn't this complicated."
Since "whacking bin Laden" is spoken of in the past tense, we can reasonably infer that this was after May 2nd, 2011 (when Osama was killed).
Since the series started on Walt's 50th birthday and ended on his 52nd birthday, the series spanned exactly two years (or more precisely 2y 1d).
(1) and (2) together imply that the series spanned at least at least 2y 30d.
This contradicts (3).
Given this contradiction, it is not possible to arrive at any conclusion regarding how much time has passed in Breaking Bad.
Q.E.D.
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I'm pretty certain that season 2 episode 12 was set in March 2009 as Jane was born in April 1982 and
her dad said she would be turning 27 the following month.
Working back using the timeline posted elsewhere the series began in October 2008 and this was when Walt turned 50.
What all that means is that each season has only covered about three months.
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Unlike some shows that generally keep similar pace with real life—where a season is a year— Breaking Bad  doesn't seem to.  Currently (part-way though season 5), the characters were remarking that it had been a year since Walt's cancer diagnosis, though I forgot when that was relative to the premiere (if it happened 'then', or in a flashback).  How much time (in universe) has passed in each season of  Bad ?
How much time has passed in Breaking Bad?
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Season 1 starts off with Walt's 50th birthday and at the end of Season 5, he starts off the day by celebrating his 52nd birthday. So, Exactly 2 years have passed by the end of the 5th season.  Exactly two years, by the end of the 5th season
 Exactly two years, by the end of the 5th season. Season 1 starts off with Walt's 50th birthday. When Walt comes back to Albuquerque in the end of Season 5, he starts of the day by celebrating his 52nd birthday:   