If it is possible for a General Aviation flight to land at Reagan National Airport, what would you have to do?
If it's not, what kind of security clearance do commercial carriers and their pilots have to go though in order to land there?
The TSA operates the DCA Access Standard Security Program (DASSP), which allows general aviation flights originating at designated gateway airports to fly into DCA, however there are many restrictions in place, some of the most significant being:
So the answer is "Yes, you can do it, but you probably don't want to."
It also makes no sense to do it in something like a Cessna 172 as you're giving up at least one seat for the security officer. The program is really geared toward larger aircraft (business jets, etc.) transporting executives with a business reason to be in DC, like testifying before Congress.