What are the nine things that the FBI looks at during a background check for a possible new employee?

Assuming you pass the initial polygraph test (very high failure rate), you can expect the following during the formal BI, which is arguably more difficult than the CIA's:

Full records check and interviews, including:

Education (minimum 4-yr degree is required to work at the Bureau)

Financial - full credit and banking records

Medical

Legal/Criminal - anything on your record, including any drug use or trouble, even minor traffic violations

Associations (groups, friends, organizations, etc.), including any overseas contacts

Employment history / problems

Expect in-depth interviews of:

Family (immediate)

Friends (local, high school, college, work, and more)

Neighbors

Employers and current/former co-workers

Current and former Teachers

The interviews are to get a more complete picture of who you are as a person, and your true character, which isn't always apparent with a records check.

To say the FBI leaves no stone unturned during a prospective employee BI is an understatement; that's why passing the polygraph is so difficult, and why the Bureau won't commit to a BI unless the prospect passes. Even then, not everyone gets through the BI process.