Gun Show Background Check Act of 2013 - Amends the Brady Handgun Violence Prevention Act to require registration of gun show promoters and to set forth the responsibilities of promoters, licensees, and other transferors.

Provides that if any part of a firearm transaction takes place at a gun show, each licensed importer, manufacturer, and dealer who transfers one or more firearms to a person who is not licensed shall, within 10 days after the transfer, submit a report of the transfer to the Attorney General. Sets forth penalties for violations.

Grants the Attorney General authority to enter the place of business of any gun show promoter and any place where a gun show is held, during business hours and without a showing of reasonable cause or a warrant, for purposes of examining records and the inventory of licensees conducting business to determine compliance with this Act.

Increases penalties for: (1) serious record-keeping violations by licensees, and (2) violations of criminal background check requirements.