The best reference I know of is in the book The Kerr Spacetime, edited by Matt Visser. David Wiltshire and Susan Scott.
The introduction by Matt Visser contains a lot of additional info on the original paper, and the subsequent chapter by Kerr contains a detailed account of everything that motivated him to look for the metric and the steps that don't appear in the original paper.
A summary of the idea is this: in the null tetrad approach start with an algebraically special metric (see Petrov Classification) and use the Goldberg-Sachs theorem to simplify the tetrad. Then assume the existence of two groups of isometry, namely that the spacetime is stationary and axisymmetric. Then impose asymptotic flatness. Write the Cartan structure equations, that with these assumptions should reduce to ODEs, the solution of which gives a two-parameter family of metrics, the Kerr family. The book spells it out in considerable detail.
You should know there is another way of getting the Kerr metric though, with the use of the Newman-Janis algorithm. The paper by Drake and Szekeres "An explanation of the Newman-Janis Algorithm" is a good starting point as any. The idea of the NJ algorithm is that by starting with a non-rotating solution of the Einstein Equation (Schwarzschild in this case) you can obtain the rotating generalization by means of a complex substitution. It seems almost magical because if gives the Kerr metric with little effort (at least comparing with Kerr's original derivation), and it works for tons of other cases too. The trade-off is that it seems obscure at first why the algorithm should work at all, but the aforementioned paper does a good job at explaining it. This method is the one Newman used to get the Kerr-Newman metric (the charged rotating case) from Reissner-Nordstrom.
