Each company will specify the checks required. Most airlines only have an engineer 'in attendance' for the first and last flights of the day so the external checks in-between are totally down to the flight crew. Basically looking for loose panels, birdstrike/ground equipment, leaks, tyre condition and any bits hanging off that should be hanging on. If you see the F/O doing it it is probably raining.
Most 'walk-rounds' begin at the steps and go clockwise so we don't get lost.