I've had 7 neuro exams preformed by 6 different neurologists, so I'm pretty qualified to give a response here.
Most start out by gathering your family history. They'll ask textbook questions like any history of strokes or bt, then go on to more specifics like MS or the sorts. After that, they gather your current symptoms, and ask questions pertaining to those. After they have gathered the necessary information, then they'll proceed to the clinical.
Almost all the clinical's I had were virtually the same.
First, they'll test your strength. They'll have you spread your fingers apart, and then they'll try to push them in. They'll have you put your elbows up to the side, and they'll try to push them down. They'll have you form the 'o.k' sign and try to pull your fingers apart. They'll force down on your shin, and tell you to push them off. After the strength is checked, they'll move on to something like coordination.
For coordination, they have you perform a couple of exercises. They'll have you tap your nose, then touch their finger in varying places in space. They'll then have you tap your forefinger to your thumb, and see if the rate of tapping is the same for both hands. Then they'll have you place your hands on a solid surface and flip flop them back and forth, kind of looking for the same thing.
For reflexes, they look for hyperreflexia. They check the knees, elbows and ankle areas, and sometimes forearms. They also perform the babinski test in which they scrape the bottom of the foot, looking for the ways in which the toes curl. If they curl up, its a sign of a lesion.
They test sensory nerves by using a vibrating tuning fork, and usually placing it at the feet. They tell you to let them know how long you feel the vibration and to let them know when you no longer feel it. They do this for both feet. They also test your sense of feel as they'll prod you with a sharp to see if you can sense the pricks, and then they'll also run a cold metal along your skin to see if you can pick that up as well.
Then they usually like to check the back of the eyes and look for any ware or tare on the optic nerve (this is usually done for MS). They'll have you follow their finger with your eyes (looking for nystagmus).
Finally, they test things like balance, and sometimes they like to look at your gait, and they'll have you walk down the hall or what not.
Every neuro has his or her own style, but for the most part, they follow the outline shown above, especially if you're being worked up for MS