Urushi is brown, amber, sometimes yellow. Black (kuro) is made by adding iron compounds.
All other colours – by adding pigments (not dyes). When adding pigments mixing and filtering is your main job.
On video below, you will see the basics of this process – mixing urushi with red pigment, controlling colour with titanium oxide, mixing tips and tricks.

Urushi used: nakanuri
Pigments: dark red (made in Japan, based on titanium oxide), pure white titanium oxide, eggshell powder.
Tools: spatulas, metal, plastic and wooden.