I have 6" ceramic tile that I would like to change the color of the grout.
There is cement board under the ceramic tiles. Can I remove the grout? Skim off part of the grout and regrout? Will the new grout adhere to the existing grout if I do not remove the entire grout?

You can remove the grout but it isn't fun. Fein, Dremel, Rockwell and others make a tool that vibrates and the Dremel has a blade made for removing grout. Tool runs about $100 up to $400 and blades aren't cheap, $25. Fein vs. Rockwell vs. Dremel (Multimaster tool) - Ceramic Tile Advice Forums - John Bridge Ceramic Tile

Is it better to remove the grout entirely or just enough to put another layer overtop of the old one?

Entirely gives a much better crack free grout. You won't get every speck of it but I'd shoot for more than 90%

Thank You,
Your Awesome !

If you buy one of the tools you can probably be fine with the Dremel for about $00 at Menards. Fein will last way longer but at 4X the cost