If you pay inheritance tax, do you still have to claim it as income on your yearly taxes and pay again?

No. An inheritance is not taxable for income tax purposes.

Sometimes. Some property is taxed at the recipient level as well such as IRA's, pensions, annuities. Other property is only taxed at the inheritance level.

Inheritance tax, more commonly known as the federal estate tax, is a tax paid by the estate of the person who dies. Technically, it is not paid by the person inheriting the money, although it typically reduces the amount of the inheritance. With regard to income taxes, most money and property acquired by inheritance is excluded from gross income (meaning it is tax free), and is not reported on an income tax return. Some types of income, however, are taxable to the person inheriting them. Some common examples include distributions from an inherited traditional IRA and unpaid wages or pension payments of the person who died. You should contact a qualified tax professional if you are uncertain as to whether inheritances that you received are taxable.