How to Create an Excel Math Computation

I'm trying to make sense of your question.
so in this part

if I have a paycheck that is $100 and $20 is taken out of it each week, my tax rate would be 20%.

you are saying you have 20% taken out for ?both?
then in this part

If $20.15 is taken out for federal, and $5.25 is taken out for state, I want to be able to compute my tax rate.

you are not telling us what your gross is. Are we to assume it is the same $100? So $20.15 is deducted by the feds (20.15%) and $5.25 is deducted by the state (5.25% of the gross) and since they both come off the gross you can add them (20.15% + 5.25% = 25.40%).
So if that is the case, then your tax rate is 25.40%. Unless New York taxes your remaining amount after the feds deduct their amount.

A1=paycheck
B1=Fed Tax
C1=State Tax
D1=B1/A1
E1=C1/A1

Thank you for the rep!
FYI, you probably ought to remember that federal taxes are indexed too. So what may be a federal tax at one income may change when the income increases. Some states also do that, I'm not sure if NY is one that does it. Then there can also be local income taxes, NY NY may be one of those locations.

Happy to help and thanks for the feedback

My suggestion was a very basic formula, and as pointed out, does not take scales into account

Great, happy to help