There are three basic categories of civil cases that can be brought against public officials: (1) under a state or federal statutory scheme that creates specific causes of action to cover particular subjects (employment, sexual harassment, public records access, vehicle pursuits, etc.); (2) under general tort law (for such things as negligence, false imprisonment, assault and battery, infliction of emotional distress, etc.); (3) or you might be sued under Title 42, section 1983, of the US Code, for alleged violations of federal statutory or constitutional rights by state and local officials acting under color of law (similar suits against federal officials are permitted by case law; see Bivens v. Six Unknown Named Agents of Fed. Bureau of Narcotics).
The potential liability of the chief of an agency under the first two categories will be controlled by statutory schemes, or by state case law. This varies by state and by cause of action. Jurisdictions often apply a rule of law called "respondeat superior," which makes the chief liable for the torts of his or her subordinate officers. But it is also common for states to legislate immunities from tort damages for particular acts and omissions, so that you don't get sued for failing to make an arrest or prevent a crime.
Because the state rules vary widely in the first two categories of cases, it isn't practical to try to discuss the potential liabilities, immunities, and defenses for the various statutory and tort claims that could be made against you. Local advisors can do the best job of that.
But the federal rules applicable to the 1983/Bivens actions are fairly uniform, and some common principles generally apply to those cases everywhere. And because the provision for attorneys' fees over and above a plaintiff's award of damages makes civil rights cases especially attractive to lawyers, they are often your biggest headache, and the biggest potential drain on your budget.
So, when can the chief be held liable in a civil rights lawsuit?