What you need
Binoculars, a phone camera, and somewhere to stand where you can see the stack against the sky. Genuinely it. Pick two or three positions around the house that give you different faces of the chimney, and use the same ones every year so the photographs are comparable. Do it in spring, after the winter has finished doing its work and before the masonry season starts, so any finding lands at the beginning of the window, not the end.
The five things to look at, top down
The cap: present, intact, screen unrusted, sitting square. The crown: any visible crack, missing material, or a surface that looks like a thin skim instead of a cast slab with an overhang. The brick and joints: faces popped off, joints recessed or sandy, anything displaced. Staining: white chalky efflorescence means water is moving through the masonry in quantity. And the line of the stack: anything leaning, bowing or separating from the house is a stop-and-call finding, not a note for next year.
What you cannot see, and why it matters
Everything inside the flue, which is where the findings that make a chimney unsafe live: cracked tiles, open joints between them, a failed liner, heat damage after a chimney fire. Also the top surface of the crown, the state of the chase cover on a factory-built system, and the flashing under the roof shingles. Those are the reasons a ground survey supplements an inspection instead of replacing one, and it is worth being clear that no amount of looking from the driveway establishes that a chimney is safe to use.
What to do with what you find
Date the photographs and keep them somewhere you will find them next year. Anything in the leaning or displaced category is a call now. Anything in the cap, crown or staining category is worth raising at your next service visit, with the photographs, since it changes what the technician looks at from the roof. And if nothing has changed since last year, that is useful information too. Stability is a finding.