There are many methods to determine the electric field but Gauss law is preferred the most. Once you see symmetric charge distribution, it becomes easier to find electric field at any point using Gauss law. When you place a charge at the center of the conductor, all the charge spreads themselves over the outer surface of the conductor leaving no charge inside. Even if you keep extra charge at the center, it will still spread on conductor's surface. The charge enclosed is zero since no charge is left inside, therefore by Gauss law, electric field inside conductor is zero.
