Toward the end of your starter planet, after you get your companion, you should have received an option to get one of three companion customization items. To use it, hit C, then click the Companion tab, and place it in the lower left hand slot on the paper doll area. Not sure about the Capital worlds, but once I got my Bounty Hunter to Balmorra (leading that there should be one on Taris, and further beyond), there was a companion appearance vendor in the Cantina. Each cost around 12,500 credits. As for what's customized, it's not much. Skin tone, makeup (if female), hair, tattoos, jewelry is about all that gets changed.