Transistor not giving enough current
Some things to watch out for here:


The TIP120 isn't a particularly fast transistor, so make sure your PWM frequency starts out low, in the 100s of Hz range.  Your LED string may also have some circuitry on it, doing its own processing.
Try it out at DC, swapping out various parts and observing with your voltmeter.

Test your LEDs (but with a current-limiting resistor). What if they are backwards?

Test your transistor, driving the base from 5V with a 1K ohm (or so) resistor.  When the LEDs turn on, note the voltages on the base and collector of the transistor.

Finally, ensure your Arduino output can actually drive the base of the transistor. Read it with the voltmeter.