Strengthening Background Checks Act of 2013

Strengthening Background Checks Act of 2013 - Amends the NICS Improvement Amendments Act of 2007 to require each state, for each fiscal year after a two-year period after this Act's enactment, to have in effect laws and policies that require it to: (1) provide to the Attorney General not less than 90% of the records relevant to a determination of whether a person is disqualified from possessing or receiving a firearm that are required to be provided under such Act for the National Instant Criminal Background Check System; and (2) provide such records to the Attorney General in the same manner, or in a manner substantially similar to, that in which such records are required to be provided by the state under such Act.

Directs the Attorney General to withhold 10% of the Edward Byrne Memorial Justice Assistance Grant funds that would otherwise be allocated to the state if the state fails to comply with such requirements. Authorizes the Attorney General to waive such withholding for a state: (1) for one year if the state provides substantial evidence that it is making a reasonable effort to comply; and (2) for additional one-year periods if the state provides substantial evidence that it is making a reasonable effort to comply and if the Attorney General withholds 5% of the amount that would otherwise be allocated to a state for each one-year period.

Authorizes appropriations for FY2012-FY2015 for grants under such Act to states and Indian tribal governments to establish or upgrade information and identification technologies for firearms eligibility determinations. Appropriates such authorized amount for the first fiscal year beginning after enactment of this Act and cancels a corresponding amount of the unobligated balances available under the Department of Justice Assets Forfeiture Fund.