The whole point of a diamond it to look impressive. Capturing this impressive look in a photo is rather tricky.

Focus

No diamond will look good out of focus, unless you just want a nice, bright reflection. Many phone cameras do have an automatically focus function, but this often does not work at very close range or with small objects. Check to see if the camera/phone has a macro function; this is designed for close range work on small objects, so it can work quite well with jewellery. Else, use manual focusing. Find a different camera if your present one is not set up for this.

Lighting.

Perhaps this is counterintuitive, but soft lighting tends to be best for diamonds. This does not mean dim lights. It means light from no single source. Avoid anything that casts a shadow. Professional photographers often use several light sources with material over them to diffuse the light. Occasionally a light from the right side angle will make a gem really sparkle. Experiment to see what looks right. Never use a flash.

Some shiny objects cause problems for photographers by reflecting their surroundings, including the photographer. This is less of a problem with the diamond itself, but setting can be an issue. Sometime the right reflection of the surrounding can create a pleasing effect; there are no rules here, except that the result must look pleasing.

Background

Black or grey cloth is standard for the background of jewellery. It works well and there is little reason to vary it; as the main point of the photo is the gemstone there is minimal artistic possibility in varying the background. Other plain backgrounds can work well, but keep things simple and minimal. The background should basically go unnoticed.

Tripod

Taking a photo of a small object like a gemstone means you are probably using a zoom. This will magnify any slight movements, make steady photos difficult. Using a tripod will prevent this problem.

Digital Cost.

With digital cameras you can take as many images as you wish and just keep the few that work out well. Use this advantage to experiment with photos.