Diamonds are notorious for being hard to photograph, one of the main reasons for this being that they have different properties under different lighting. Non-directional soft lighting is usually found to be the best kind of light for photographing diamonds, and can be had under diffuse lights in the indoors.
Another thing you need to ensure is that everything is perfectly level. When taking pictures of a single diamond, you will need a tripod, as well as to check that the camera lens and the diamond are parallel to each other. The lens will also need to be centered on the diamond.
Note that even the slightest tilt can spoil the light performance. It will take a lot of adjustment to get the camera lens and the diamond level, so it is important to be patient.