You’ll need to have the skills necessary to become a special agent in the FBI. Background investigations are performed by agents, not non-sworn employees. There is no “background check department.” The recruitment and human resources offices may coordinate background checks, as they may involve multiple locations, but the interviews and such are assigned to special agents in the offices nearest to where the person(s) to be interviewed are located.
Some background investigation for other government jobs are farmed out to private contractors. The contractors hire investigators to visit and interview the people with knowledge of the applicant’s history. Most of those investigators are former federal special agents or police detectives.