If you are using the kit lens, zoom out to 55mm. Use P or A mode to select the widest aperture. With that lens it will be f5.6, which will have a fairly shallow depth of field at close range, but not super shallow. If you want a portrait lens with a shallower depth of field you can buy the Micro-Nikkor AF-S 60mm f/2.8.
References :
http://www.photozone.de/nikon–nikkor-aps-c-lens-tests/394-nikkor_60_28

I am assuming that you currently own the kit lens that came with your camera. Here is how you blur the background of your portrait shots, step by step:
1. Set your mode dial to "A"
2. Zoom your lens to maximum telephoto (which in your case is probably 55mm)
3. Set the aperture the lowest value (5.6 with the kit lens)
4. Keep your subject relatively close to fill the frame, and the background as far as possible, and shoot!

The kit lens you probably have currently isn’t really the best lens in the world to get you beautiful defocused background, because it’s max aperture is only f/5.6. Ideally, you want a lens with the low aperture value. If you have money to burn, get the Nikkor 50mm f/1.4 AFS, which would be able to give you excellent defocused backgrounds. Alternatively, you can opt for the (much) cheaper Nikkor 50mm f/1.8 AF, however, this lens will not be able to autofocus on your D60.
References :