Cannot get at this
There still has to be a common ground from the Arduino to the ULN2803.
Do you understand how the current will flow in this schematic?
Show us a link to the relay you are using.
Do you think this is the same?http://www.aliexpress.com/store/product/2-Channel-Relay-Module-5V-for-PIC-ARM-AVR-DSP-wholesale-and-retail/707291_499450487.html
If you only have 4 realys to drive on the Adruino, don't use the 2803.
jumper JD-Vcc to VCC, Arduino +5 to relay VCC, arduino GND to relay GND, Arduino o/p pin to IN1 and IN2.
Connect the 2803 o/p to IN1 or IN2 then External +5V (not the Arduino +5) to Relay VCC/JD-VCC.In this case a HIGH on the o/p of the Adruino will close the contacts.Of course Arduino GND to relay GND to external supply GND.With 10 relays use the 2803.