You normally use a microcontroller to multiply digital signals and then also do whatever else you need to do with the number inside the microcontroller (after all, you need a digital system to work with digital data). You also need the microcontroller to operate the ADC chip (unless you are using a microcontroller with an ADC built into it).

It depends upon the signal frequency. Remember that the A/D sample frequency needs to be at least two times the highest signal frequency (Nyquist limit) for proper sampling of the signal. And you will likely need good, high-order filters to sharply roll off the signal above 1/2 the sample frequency at the A/D input to minimize aliased noise. (The higher the sample frequency, the less stringent the filter requirements for a given signal frequency since you start the rolloff at just above the highest signal frequency).

For high frequency sample rates you can use digital multiply chips, or a digital signal processor to do the multiplication (some digital processors have built-in A/D converters). For lower frequency rates you may be able to just use a microprocessor to do the multiplication.