I recently became my brother's court-appointed guardian. He is in a psychiatric hospital right now but had a part-time job last year. Can I use TurboTax to do his taxes?

Yes, you can use TurboTax to file his taxes as his court-appointed guardian. 

Thank you!!!

How do I start that on Turbo Tax?  By entering the information of the person I am Guardian over?  Who signs the form?  Do I sign, and is there a place for me to designate that I am the Guardian?

You will sign the electronic signature itself as if you were doing your own tax return. After you e-file, you will then need to mail in the Form 8453 along with either the Form 2848 or a copy of your official POA that establishes your authority to have signed on behalf of the individual. 

Turbo Tax will provide a form 8453 for you to mail a copy of your POA along with the mailing instructions.

How to sign a return with a Power of Attorney