What makes my pixels unsharp?

I'm using a Nikon D3200 with the stock 18-105 mm lens. I have noticed that I can't focus really precisely. If I take a picture of a structured surface at an angle, I can clearly see the limits of the depth of field, but even at its centre, the pixels are not really sharp, I can see a circle of confusion of about 5 px at the full resolution. See the image (F/5.6, ISO 100, 2s, f=66 mm, cropped):

Is that something a prime lens would solve, or a property of the chip?

I think your confused. Sharpness is defined by 4 lines per mm on paper viewed from 12 inches away.

Remember that no digital camera can take a sharp picture, as the photo sites are discrete units and not one smooth surface, so a photon can go right onto a sensor bit, or it may hit the dividing line. If it hits the dividing line, its not going to register at that point. Also in play is that your camera has a anti- alias filter on it, this blends everything just enough to prevent moire patterns from showing up.

All digital images require sharpening. The amount is the trick.

Also remember that nothing looks sharp when pixel peeping at 400%. How does it look on screen or in print?