Instructions
Select your desired license and number of players and choose "Time Trial."
Choose your character and kart or bike. This is especially important when trying to unlock expert staff ghosts, since you want the most compatible combination for the track you are attempting to master. A number of players on community forums agree that Princess Peach on the Mach Bike seems to be the most effective winning combination.
Select the course you wish to run and when prompted, select "Watch Replay" to observe the movements of the normal staff ghost. You will then get an idea of how the ghost will perform. Pay attention to any shortcuts it may take.
Select "Race this ghost" after you have finished watching the replay.
Beat the normal staff ghost's time at each track by a significant margin to unlock the expert staff ghost. For example, the normal staff ghost at the Luigi Circuit finishes in 1:29:670. You must finish with a time of 1:26:376 in order to unlock the expert staff ghost. The Moo Moo Meadows ghost finishes in 1:37:856 but you need a time of 1:34.159 to unlock the expert. For a complete list of tracks and margin-of-victory times, see the expert staff ghost thread at the GameSpot forum for "Mario Kart."