Can I claim what I paid in rent on my Federal or State return?

There is not a rent deduction or credit on your Federal return.  If your state has anything for renters you will be prompted to enter your rent info when you complete your state return.  As far as I know, the states that have anything for rent are Michigan, Maryland, Massachusetts, Minnesota, Missouri, New Jersey, Rhode Island, California, Indiana,  Arizona, and Connecticut.  I don't know if there are any others.

Unless you are a business, you cannot claim rent on your federal return.  It is your personal residence and the rent is not deductible.