1.:
The GND-Line of the Power Supply and the GND-Line of the Arduino must have a connection (but only the gnd!
2.: don't use a PC Power unit: It often makes troubles because PC-Power units have their own logic. It would be better to buy a seperate 5v power supply, then it should work, if your code is ok.
If you want to see of your code is working: take the 1 Pixel you cut off your stripe and hook only this one to the arduino (directly, without power supply by a PC-Supply). Sry for my worse English ;-)
Have fun