The dyes I use are known as "Fiber Reactive," and are sometimes referred to as "MX" dyes. They are THE dyes for coloring cotton, and are doubtless the same dyes used by DMC and other floss manufacturers, as well as by popular small manufacturers of overdyed floss. They are also the dyes used by quilters to do so-called "Low Water Immersion Dyeing" of quilt fabrics.
These dyes are relatively inexpensive, easily procured, completely mixable (so you don't have to buy a thousand colors -- you can mix them!), and easy and safe to use.
Two of my favorite online vendors are Pro Chemical and Dye, in Massachusetts, and Dharma Trading near San Francisco, California. Both of these excellent companies have extensive instruction on their website for the use of these dyes.