On-the-Job Training Act of 2013 - Amends the Workforce Investment Act of 1998 to authorize the Secretary of Labor to: (1) make discretionary grants to states, local governments and boards, Indian tribes, institutions of higher education, business trade associations, and local educational agencies for adult on-the-job training or dislocated worker on-the-job training programs; and (2) use up to 10% of funds authorized under this Act to make grants to such entities for developing on-the-job training programs, including providing capacity building activities for local staff.