There appears to be no popular biography of Rush, a man who would be of interest even if he had not been a signer of the Declaration of Independence.
As our readers know, Rush was the preeminent American physician of his day, the founder of American psychiatry, a pioneer in the humane treatment of the mentally ill and of prisoners, a pioneer of the therapeutic approach to addiction - and an ardent revolutionary who antagonized Washington (he wanted Washington out as being an inadequate strategist - which he was).
He was also a passionate Christian whose faith guided his life, and his fierce opposition to slavery did not endear him to many. He founded the first Bible society in the US.
This book by Brodsky seems like less of a popular bio than a compilation of Rush's huge correspondences. 2000 Rush letters survive.
This one by David Barton might be the best one out there.
Here's Rush's autobiography
Here's his book, Medical Inquiries and Observations Upon the Diseases of the Mind.