How should I wire up ground when using an ATX power supply to power an Arduino and several LED strips which in return are controlled by the Arduino?
Since you are using the same power supply for both, the Arduino and the LEDs, you do not need to connect the ground pins -- as long as ground is connected internally. 

As you have recognized this would be redundant and unnecessary and could even lead to further problems (there would be several ground loops). So I think connecting the Arduino to ground just one time will be fine.