Use a serial lcd and connect it in parallel with the Pins.
If data sent to the PC application is binary, you will not be able to display/view the data on the LCD. The binary data may even reconfigure your display (switch it off, change baudrate, enable blinking, change backlight etc.). Even with ASCII only (e.g. GPS data) it is not practical because the volume of data is likely to overflow the LCD display quickly.
Just make sure your debug statements include a \n to jump down to the next line on the LCD.
Anything you add to the serial stream will also reach the PC application and may upset whatever protocol you're implementing.