that means that computer connected to RX can use RS232
Not sure what you mean by this.
The computer is normally connected through th USB, not RS232.
As Sacman says you need a level converter if you are going to use RS232, something like the MAX202.
The USB to serial converter in the arduino is connected to the chip through 1K seriese resistors, so any signal feeding into the RX pin will have to be strong enough to over ride this.
On the T X side the signal will go to both your display and the computer unless you make some hardware mods.
You need both RX and TX free to upload a sketch.