We sharpen our own chains using the Diamond stones below in a Dremmel. It makes razor sharp chains for ice sculpting.

First, unplug your saw & clamp it to a work table. (I like to work about 40″ off the ground)
View of the chain
Use a marker to mark a tooth, so you can easily tell where you started

Sharpen all of the teeth facing right

Using a dremmel or Mini Die Grinder with a diamond sharpening stone, gently rub forward and back against the tooth.

Using your eye, keep the same angle.

Be careful to not let the sharpening stone jump over the front of the tooth

Repeat this process for all teeth facing left