No. You cannot claim the additional $3000 if you are already claiming the amount of FSA. The IRS does not allow double-deduction. You can utilize either the Dependent Care FSA or take the Dependent Care Tax Credit, but you cannot take a tax credit for expenses that have been reimbursed by the FSA.
To determine the available credit, you can consider up to $3000/annually of child care expenses for one dependent, and a maximum of $6000/annually for 2 or more dependents. Depending on your adjusted gross income, the maximum credit (dollar-for-dollar offset of tax liability) is:
In other words, the higher your earnings, the lower your tax credit
In general, those with less than a 15% tax bracket will be better served by the Dependent Care Tax Credit. IRS Publication 503 "Child and Dependent Care Expenses" provides full information about this tax credit and offers worksheets and aids for performing the calculations.
Specifically:
The $3000 was out of pocket expense and not reimbursed by the FSA. I had the $5000 in FSA and had to spend $3000 out of pocket. I want to know about the out of pocket expense. Do I claim any of that or not?
Unfortunately you cannot because you are already claiming the FSA, and you have only one child. The maximum you can claim for a child is $3000. If you had 2 childs, you could have claimed additional $1000 from the out of pocket expense.