Sending multiple electrical signal to PC via Serial port
If you already have an ADC, you will need something to read your ADC!

You mention a serial to USB cable in your OP. Let's assume that the task is modest enough for all the information to fit into a standard serial data stream.

Without wishing to plug any particular manufacturer, I will tell you what I do for the simplest possible solution of hooking something dumb and digital to a PC.

Use an Arduino (Uno is the easiest to start with) which plugs into the PC USB port directly. The USB allows you to program it. It also allows the PC to communicate with it using standard serial protocols, and is fast enough to talk serial and then do some extra work. The standard Arduino support software on the PC contains a monitor to allow you to talk with it, for testing purposes. You can then use any programming language running on the PC (Python with the 'serial' library is my choice, but anything else you're comfortable with will do, and if you're not yet comfortable with anything else, then Python is my recommendation for what to learn) to talk with the serial port. You can also pipe files to/from the operating system. 

Write a program on the Arduino, in C, to read the ADC, format the data, and send the data to the PC. Use parallel, SPI, i2c, many standard protocols have libraries to simplify the task.

You can get other Arduino boards that have a serial interface, and use a USB to serial converter or cable for interface. Although the Uno plugs in directly, it does use serial under the hood, so they are more equivalent than it appears at first glance.

I seem to have used the word 'standard' a lot in this answer. That's because you won't really have to invent anything to do what you want, just hook up standard components and protocols. The tricky bit is making sure you understand what each has been designed to achieve, and then using them that way. 'Ride the horse in the direction it's going'. There's nothing more frustrating than trying to use a tool or component or library to do something it's not been designed to do.

Of course if you are already versed in PIC, then use that instead of Arduino. But I guess if you already use PIC, you wouldn't be asking the question.