If you inherite money, after the estate taxes are paid, do you have to claim income on the balance that you get? What about in NJ?

The answer is no. The estate taxes are taxes on the money in the estate and it is taxed at the time of the disbursement of the estate. If you inherit things like stocks and bonds, your new basis in them will be the value on the day you inherited them in case you sell them. The inheritance tax is only going to come into effect on estates over a certain size, so unless the estate is over that number there will be no taxes. The executor of the estate should supply you will any information that you need.