I'm not a terribly experienced coach, but I've spent about 5 hours a day, 6 or 7 days a week in the gym for the last 10 months helping bodybuilders, crossfitters, and jarheads learn how to pick up weights without breaking themselves.
During the first lift I look for obvious deficiencies. Things like the athlete chasing the bar, getting all wobbly at the bottom of the snatch, feet landing mega-wide, blatant bar path eccentricities. If the lifter doesn't seem to have any obvious flaws, then I break the observation down further like Greg was just saying.
Having made it this far, the athlete would have to have a pretty decent grasp of the lift. For me personally, it's easier to watch a section of the body than it is to try to break down the lift into its technical components when I'm looking for problems. It just requires entirely too much concentration for my ADD ass to follow.
I start at the bottom watching the feet and shins, then the midsection (back and hips), then the bar path, then the elbows in the clean and the shoulders in the snatch. It's worked pretty well for me so far...no one has left the gym more broken than they were when they came in, though an awful lot leave with a newfound hatred for PVC pipe.