You will need a camera with a macro lens. Compact cameras often have a marco mode, but this can vary widely. It is normally specified as the closest focus distance the camera is capable of. 1cm is good, 30cm means your coin will need to be the size of a plate. Ideally choose a camera that gives some manual control to settings like aperture and exposure compensation.
If you use an SLR the macro capability is specified as a ratio. Most macro lenses have a ratio of 1:1 meaning it can focus on an object the size of the cameras sensor and the object will fill the resulting image. If your SLR has a full frame sensor aim for a lens of about 90mm. If your camera has an APS size sensor aim for a lens of about 60mm.
You will need a good light source for macro photography. A table light can be used but choose one that provides lots of light and is easily positionable. Table lamps can give a orange light compared to daylight which gives the photographs an orange tint which will require the cameras colour balance to be set to compensate or the photos to be adjusted later on your computer. A daylight balanced light source is a better choice but photography lamps are not cheap. Position the light source so the light appears to be from above in the photo. Having the light appear to be shining from under the chin of whoever is on the coin will look unnatural.
A flash can be used but most macro flashes are the 'ring flash' type. These are not very good for photographing coins as they give an even light from all directions. This is undesirable as shadows help accentuate detail. Choose a macro flash that has adjustable flash heads or a ring with several segments that can be individually controlled or disabled.
The best background is a light box. This provides an even white background resulting in an image of the coin with a white background and no shadows. However it can be difficult to balance the exposure so the coin is well exposed and the background is overexposed to pure white especially with compact cameras. Best results require a slow shutter speed to ensure the background is overexposed and the camera exposing for the coin and not the background which requires a camera capable of spot metering to correctly expose the centre of the image rather than set an average exposure for the whole image.
An alternate is to use a plain coloured background. Choose a colour that contrasts the coin. Black is good for silver and gold. Red or blue is good for bronze or dark silver coins. The background should be of a material that is non reflective so the light source does not produce very bright areas. Cotton velvet seems to be the best I have found. Be aware that most velvet is made of synthetic materials and plastic can be very reflective. You will get a better picture if the coin is not lying directly on the background so the background becomes slightly out of focus when the camera focuses on the coin. An easy way to achieve this is to put the coin on a small stack of dice.
The best camera mount I have found for macro photography is an Opticron hide mount. It is designed to clamp a camera or telescope on the side of a hide for birdwatching but works well clamped to a table with the camera pointing directly downwards. A tripod can be used but can be difficult to position without the legs either being in the image or casting shadows.