This answer contains some additional resources that may be useful. Please note that answers which simply list resources but provide no details are strongly discouraged by the site's policy on resource recommendation questions. This answer is left here to contain additional links that do not yet have commentary.
The original reference is Kerr, R.P. Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11, 237 (1963).
you can also read this paper , it is very useful: A Possible Intuitive Derivation of the Kerr Metric in Orthogonal Form Based On Ellipsoidal Metric Ansatz.
I'm surprised no one mentioned Chapter 6 of Chandrasekhar's Mathematical Theory of Black Holes. Then again, I personally found his derivation so opaque that I wrote my own de novo derivation in Boyer-Lindquist coordinates using the Cartan formalism.
I think if we follow the Ray D' Inverno book intitled Introducing Einstein Relativity then we get easily this metric. on chapter 19 it is explanied in a very simple method
Here is a review of the process of discovery by the man himself: icra.it/icranet_report/08Kerr.pdf The OP might like to read this and see if he still has the fortitude for the task!
