Arduino Uno & Bluetooth module

Hi everyone, french student here, i'm trying to add a bluetooth module to my Arduino Uno, but it seems that the connection between the Arduino and the module doesn't work.
My PC and the module are paired : the led stops blinking when I open the monitor, and I can use the cmd mode.

However, I can't receive/send data to and from the Arduino : I don't get anything on the serial monitor.
The module is this one.

The wiring looks good :
Arduino Module
3.3V ------ VCC
RX -------- TX
TX -------- RX
GND ------ GND

And I'm using this code :

void setup()
{
  pinMode(13, OUTPUT);    
  Serial.begin(115200); //Initialize serial at 115200bps
}

void loop()
{
   while (Serial.available() == 0); // wait for character to arrive
   char c = Serial.read();
   if(c =='1') {
     digitalWrite(13, HIGH);   // set the LED on
   }
   if(c == '0'){
     digitalWrite(13, LOW);  // set the LED off
   }
   delay(500);
}

I really don't see where the problem comes from, any idea ?

Has your bluetooth module been configured for 115200.? I bet it hasn't been. If not change the code to 9600. It is more likely to work.

Are you sure 3.3v is correct for Vcc? If not, 3.3v might not stop it working, but it is still a bad idea.

What device are you using to communicate via bluetooth? If it is the same PC which has the serial monitor, it won't work. Use a phone or another laptop. You can use the serial monitor to see what is happening but not to send. This is because monitor and bluetooth share the same port.

Thanks for your answer,

Nick_Pyner:
Has your bluetooth module been configured for 115200.? I bet it hasn't been. If not change the code to 9600. It is more likely to work.

I tried changing the code to 9600 : didn't work. I also tried with 115200 in the code AND on the bluetooth device

Nick_Pyner:
Are you sure 3.3v is correct for Vcc? If not, 3.3v might not stop it working, but it is still a bad idea.

Yes I'm sure, user manual says so (PIN 1: 3.3V regulated power input to the module)

Nick_Pyner:
What device are you using to communicate via bluetooth? If it is the same PC which has the serial monitor, it won't work. Use a phone or another laptop. You can use the serial monitor to see what is happening but not to send. This is because monitor and bluetooth share the same port.

I tried using both the same PC which has the serial monitor, and an android phone, didn't work for both of them. I don't receive anything (even if I add a Serial.write("test") at the beginning of the loop)

Could it be an electrical problem ? The TX and RX leds on the Arduino don't blink at all with the module

You might find the following background notes useful

http://homepages.ihug.com.au/~npyner/Arduino/GUIDE_2BT.pdf
http://homepages.ihug.com.au/~npyner/Arduino/BT_2_WAY.ino