Would you be able to use a car alternator to produce electricity and hook it up to your house grid simultaneously well receiving electricity from the local electric company?

Well... not only with an alternator. Power is produced by a generator. An alternator loads the generator output into a battery bank.

You need additional electronic equipment to change the power in that battery bank into AC at 110-120 or 220-240 Volt , and in phase with the net supply. Available from solar panel system suppliers.

You need to make arrangements with your energy supplier prior to delivering energy to the grid.

Just one question : how do you intend to power that generator? With the current petrol/diesel prices you will loose money ! Do you perhaps have a windmill, a water wheel, or a solar panel system ?

Success !

:)