If you do the voice recognition on the PC and the voice recognition software can send data over serial to Arduino then that's fine, but you still have the problem with controlling the browser. If your voice recognition software can't do that you have a problem.
The serial port can only be used by one program at a time, so if the voice recognition software is using the port GoBetwino or anythig else can't use it.
So baiscally you need ONE piece of software on the PC side that can do all the things you need that involves serial communication with Arduino.