What paperwork do I need to sell a car for $30k cash in New Jersey and deposit the proceeds into my bank account?

You don't need any paperwork other than a deposit slip to deposit the cash into the account. The BANK will need to fill out a form to help the Government detect money laundering, counterfeiting and other crimes. If you aren't doing any of these things you have nothing to worry about.

This is not going to trigger any tax liability unless you report it on your tax returns as a capital gain or other type of income.

The one thing you DON'T want to do is split it into several smaller deposits that are just below the reporting limit. That is called "structuring" and will trigger some alarm bells you don't want rung.

Your main risk is being tricked by the buyer. There are not many (legal) reasons why someone has 30k in cash but cannot use bank account to pay you, so chances are the money is illegal. That is not really your problem, unless the police thinks you should have reasonably known that, or the money is fake.

If the government comes after you (chances are they have bigger fish to fry), you should be able to prove how you came into posession of it - which should be easy, the car ownership was in your name, and is now transferred. Having a copy of the sale contract might be helpful.

It is legal to sell your car for cash; just be careful that you don't get tricked or robbed.