How to transmit data from an analog multiplexer to an RS232 COM port on a PC
My understanding is that an analog multiplexer can be used to combine
  the voltage signals and transmit them to the computer's com port


An analog multiplexer is just the electronic equivalent a multi-pole rotary switch - it simply steers one of the inputs to the output. The output is still analog, not suitable for transmitting via RS232 (which is a +-12V digital signal).

To create the serial signal you need an ADC (Analog to Digital Converter) to convert the analog voltages into digital numbers, and a UART (Universal Asynchronous Receiver Transmitter) to serialize the data bits with the correct baud rate and format. 

You could make such a circuit using separate analog MUX, ADC and UART (with a few logic gates to 'glue' them together) or just use a small microcontroller which has all these features built in. 

An MCU might seem more complex because it needs software, but the hardware can be much simpler. You will be sending 12 readings to the PC continuously and it needs to know which one is which, so your transmission protocol must include more than just the raw data. You will find it much easier to develop this protocol in software, rather than having to add even more chips to a complex digital control circuit.