misterinnovation:
I would like to be able to set it with my computer through the usb cable. [like pressing "S" to display what time the alarm clock is set to go off, then "H" to increment the hour counter up, and "M" to increment the minute counter up.
Yes, it's entirely feasible and very straight forward to do. You need to use an application on the PC to receive your keyboard actions and send the corresponding characters to the Arduino via the USB connection. The Arduino serial monitor provides this capability, or there are various other applications you could use.
On the Arduino side you'd need to write a sketch that read from the serial port and parsed the received characters to work out what command you were sending. Output from the Arduino follows the same path back to your PC for display.