These notes are too pedagogical to be published, but I wrote them up while starting new projects. Please let me know if you find any typos or errors!
This is essentially Chapter 1 of Chandrasekhar's book ''The Mathematical Theory of Black holes'', some notes on the GHP formalism, and notes on different coordinates and tetrads for the Kerr spacetime. I also wrote a mathematica that derives the Newman-Penrose equations, and goes through a lot of different coordinate systems for the Kerr spacetime as well.
Notes I have written up while working on a few projects. There are a few nice articles on this subject (which are cited in the notes), but I wanted to redo the calculations to double check.