I am trying to build a standalone Arduino. For certain I am not getting that USB to Serial breakout board(SparkFun FT232 breakout board). Is it possible to directly connect rx tx pins etc of arduino to rx tx of serial port of the computer attached directly to motherboard.
No. The PC uses RS232 signals (-12V/+12V) and the Arduino use TTL signals (5V/0V). You need a level conversion. See the schematic of the "Single Sided Serial" version of the Arduino for an example of how to do the level conversion with resistors, transistors, and diodes: