Pigments are insoluble and impart colour by reflecting and imparting visible light. Okay enough chemistry. Dyes therefore have the advantage of penetrating the fiber, resulting in a somewhat more flexible and translucent sheet than paper colored with pigment. However, dyed paper will generally fade over time as the chemical elements continue to react to sunlight and oxygen. Natural dyes can be made by crushing plant materials, soaking overnight, boiling, straining, then adding fiber to the colored liquid.
Dyes are also available commercially.
I've experimented with adding powder cold water dyes to my pulp before adding the pulp to the vat. Pigments are the colorant of choice for many papermakers. Synthetic pigments have similar properties. Professional pigments available from papermaking suppliers come with suggested formulas for the amount of pigment and retention agent to use based on the quantity of dry pulp. When properly pigmented, the pulp should be colored, and the water should remain clear.