How can I make this costume mouth open and close?
As a basic strategy, you might consider having the elastic/spring hold the mouth open, and use something controlled by your mouth to close it when you close your own mouth. That way you could make him talk in a realistic way by simply opening your mouth each time you say a word, which of course you'd want to do anyway :)

As far as mechanism is concerned, a "string" (tensile member) of some sort might be easiest. My mind goes to a cable of the sort used on bicycle hand-brakes, where the cable runs through a housing. This would give you flexibility to position the actuator anywhere you wanted without having to worry about keeping the "cable" tight. The cables they use for gear shifters are lighter gauge and would still be plenty strong enough. Maybe you could adapt a set of costume teeth (maybe vampire type or something) to act as the actuator. That would fit in your mouth but still allow you to speak and you could maneuver them in and out of your mouth I expect without too much trouble, even without use of your hands.

Maybe the "mouth controller" end of the cable could just pull the bottom lip in when you close the teeth, perhaps with a coil spring (ball-point pen type, maybe longer) doing the needful to open the mouth on the return stroke.