Should I claim my rent if I'm not on the lease

RENT

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 Arizona, California, Connecticut, Hawaii, Indiana, Iowa, Maine, Maryland, Massachusetts, Michigan, Minnesota, Missouri, New Jersey, New York, Rhode Island, Vermont, Washington DC, and Wisconsin.

https://www.rent.com/blog/states-with-a-renters-tax-credit/

and each of those states has different requirements, so check your particular state's rules.