So do have a bicolour LED? That's one with two leads, and a pair of LED dies inside it. Connect it one way round it glows red, the other way round it glows green.
Or maybe a tricolour LED? That's one with three legs. One common, one leg for red and another for green. If you connect a single colour and common with the correct polarity it glows that colour, and if you connect both colours with seperate resistors, the colours mix to give yellow.
If its a common cathode tricolour LED, its easy, just use the circuit you posted. Common anode is harder, either you have to rebuild it with PNP transistors so you can reverse the supply to the whole circuit, or you have to add two extra transistors in place of the LEDs above to drive a common anode tricolour LED.
Bicolour LEDs also need the extra transistors, but the details of the wiring are slightly different, and the simple circuit wastes more power.
We'll get to the circuits when you tell us how many legs your LED actually has, and if its got three legs, whether its common anode or common cathode.