There have been 70,291,049 background checks for gun purchases since President Obama took office, according to data released by the FBI.
In 2009, the FBI conducted 14,033,824 background checks. If we subtract the month of January (Obama did not assume office until the end of the month) we get 12,819,939.
The FBI conducted 14,409,616 background checks in 2010, 16,454,951 in 2011, and 19,592,303 in 2012.
Add to that the first three months of 2013 (2,495,440, 2,309,393 and 2,209,407, respectively) and the total number of background checks under President Obama comes to 70,291,049.
Over the same time period, the number of background checks completed under President George W. Bush was 36,090,415, or about half the number conducted under Obama.