A real classic B movie is something made for the drive-ins back in the 1950s through late-60's. Mostly sci-fi (like Plan 9 From Outer Space previously discussed-the stuff you now see on MST3000), some horror, some monster (like a zillion Godzilla movies and its clones), motorcycle gang movies (best if Jack Nicholson is in some minor role). In the mid-to late sixties drug and hippie movies were a fertile B movie subject, like The Trip by Roger Corman starring Jack Nicholson.
Since drive-ins are nearly nonexistent anymore (only two in my state, and one of those is part time) the only real way to watch a B movie now is on some grade Z TV station with low budget commercials.
And yes, Dr. Strangelove is definitely NOT a grade B movie, and shouldn't be considered one just because it was done in black and white. Stanley Kubrick directed it, one of the greatest directors ever.