Start the single player Campaign mode under the Regular or higher difficulty level. Complete the "Ghost Stories", "Brave New World", "No Man's Land", and "Struck Down" missions to unlock the "Extinction Mode" option at the main menu. Alternately, start Multiplayer Career mode and reach level 5.
Golden Aliens in Extinction
Anytime during Extinction mode, head over to Gil's Lodge Motel and shoot the letters L, O, L on the motel sign to trigger Golden Aliens for the next 90 seconds. You can repeat this as much as you want.