How to produce photos like these?

I want to take close-up photos of insects like these:

I have a nikon D3100 camera and a Nikon 60 mm macro lens, but I can't get extreme close ups of insects. What would be best to do to get photos like these with my current lens, because I don't want to spend money on a whole new lens? I don't think that extension tubes will work well because my minimum focus distance is already about 20 cm and any closer will block the light and scare the insects away.

Also what Nikon lens would take photos like these?

You need macro lenses for this purpose. Check with the official nikon website to find out what lenses they have.

So you have the D3100 and use the macro lens? I'm planning of getting that too. Maybe your technique is wrong?

Macro lens and extension tubes…

or kit lens with a reversing ring…

Using a longer focal length of lens will put you further away, a 100mm macro lens would put you about 5" (13cm) away at 1:1 on your camera, a 60mm lens will be (a lot) closer than that around 5 cm.

You can try a set of close up filters on any telephoto lens you have, cheap to try, image quality won't match the dedicated macro lens, but it's better than you might think for something so cheap.

I use an old 135mm manual lens on a set of bellows for my 'bug hunting kit', at 1:1 I'm about 18" away, well out of the insects 'radar' zone, then I can easily get shots like this. Altogether a cheap bit of kit which I much prefer to my Macro lens as I can set the magnification, higher than 1:1 if I want.

There's a downside, the further you are away from the insect the more you magnify any camera movement, I get around this by using flash. A manual flash on low power which is enough even with the small apertures you have to use to get any depth of field, has a flash duration around the 1/50, 000th of a second mark which freezes any camera or subject movement. The only problem you have then is focusing which at these magnifications is best achieved by moving the camera/lens contraption backwards and forwards to achieve focus, auto focus is pretty useless as depth of field is so small. Live view helps loads with focusing.