how do i sue a citys insurance company

Although I do not practice in North Carolina, the general rule is that all or most cities (i.e. municipalities) carry liability insurance to cover these types of matters.  However, filing suit against municipalities can be more complicated than doing so against a private citizen and/or corporation, in part due to the fact that additional notice requirements may be necessary when the defendant is a municipality.  To answer your specific questions, anytime an individual receives injuries due to the conduct or omissions of someone (or something) else, that conduct/omission may make the entity at fault liable in a personal injury/tort action.  Personal injury attorneys usually work on a contingency basis, which means that they only get paid out of a percentage of the money recovered on your behalf.  What this means, in practical terms, is that very little or no money is often required by you to hire this type of attorney and/or recover money for your injuries.  Therefore, I recommend that you consult with and/or retain a personal injury attorney to evaluate the strengths of your claim and to represent your family's interests with respect to this potential claim.