How does a US registered pilot obtain approval to fly to Cuba?
To obtain approval to fly to Cuba you must first sumbit an application for an OFAC (Office of Foreign Assets Control) License through their Cuba Travel website or by mail:

Office of Foreign Assets Control
U.S. Department of the Treasury
Treasury Annex
1500 Pennsylvania Avenue, NW
Washington, DC 20220

Once you have your OFAC license, that completes your US legal requirements, as far as I'm aware.* There are other requirements that you have to fulfill to receive a Cuban visa upon entering, but since you have asked about obtaining US approval I haven't looked into that in detail.
For the record, this is the same method that anyone would use to travel to Cuba. Pilots flying to Cuba must ensure that all their passengers also have their own individual OFAC license.

* I am not a lawyer. Please do not consider this to be legal advice. I would recommend calling OFAC and inquiring for further help with the process.

Sources:

Cuba: What You Need To Know About U.S. Sanctions Against Cuba - OFAC
OFAC Contact Info