Sorry for all the fear mongers here. Ignore them, most are couch surfers. What you need to do is apply for a TIP on line as Rob mentioned. you will need the original registration for the vehicle and you will get a 180 day permit. Here is the link you need to follow.
TIP on lineDo not forget to cancel it when you exit Mexico. You cannot do it at all border crossings.
Your big issue may be that you can only apply within 60 days of the permit being valid. that may be an issue if timing means you have to bring the car in with some of the 180 days already expired.That is legal but will cut down on your time there.
As far as insurance is concerned, there are many companies. Mexpro is one of them, also Lewis & Lewis, Don Smith, among others. Shop around for the one that suits you best. Make sure you have a legal rider (they provide a lawyer) and that you have $500k liability.
You will have to pay a deposit on the car ranging form $200-$400 depending on its year. You have to use a credit card or US dollars and you will get it back when the vehicle leaves Mexico.
BTW on edit regarding the fear mongering. Statistically an American citizen has twice the chance of being murdered in his own country than he does in Mexico.(
Link) A higher proportion of Canadian tourists (based on visit-nights) are murdered in Australia than they are in Mexico.(
Link) Most murders in Mexico are targeted. Unless you are Mexico to buy drugs, your risk is low. Your risk of being a victim of property crime is higher, however. The murder rates in the 2 largest cities, Mexico City & Guadalajara, are lower than about 40 American cities. Just stay out of Cuidad Juarez
Oh, and don't turn right on a red, and be aware that your left hand turn signal means its OK to pass you to the guy behind. if your car has the drive on the other side, be prepared to get shaken down by a transito who may tell you its illegal. Maybe it is, I would check.