Step One - you can't even begin to think about taking a photo if your ring is not clean. So make sure it's clean and buff the face of the diamond with a soft cloth to take care of any fingerprints before you even get started.
Step Two - get the light right. A lot of people make the mistake of going for really bright light and that actually washes the diamond out in a photo and it is hard to capture in a still photo. What you actually want is diffused light. What does that mean? Think about taking a photo when the sun is shining but you have blinds pulled down, diffusing the light that streaming towards your diamond is the best way to capture the natural sparkle.
Step Three - a lot of the fabulous photos you see on Instagram (mine included) are taken with what's called a macro. A macro is a lens you clip onto your phone and it magnifies the image, making it easier to capture the design and the diamond. You can easily buy a basic one on Amazon for about $20.
Step Four - find a good background. My trick, is I actually buy paper and wrapping paper from stores like Paper Source and use those as my backgrounds. You don't have to that but it is a handy trick for making should the background always looks nice.