I want to use pure C to read and write to the serial port of my computer from an arduino uno.
Um, go ahead!
Starting with the older Arduino code (HardwareSerial.c) might be helpful, and I'm pretty sure there are lots of examples over on AVRFreaks.
http://www.avrfreaks.net/index.php?name=PNphpBB2&file=viewtopic&t=48188
http://www.avrfreaks.net/index.php?name=PNphpBB2&file=viewtopic&t=45341