If you know the commercial names of the colorants, or even better their Colour Index designations, then you may be able obtain their molecular structure from a copy of the Colour Index - see link for details. The CI is available online, although is a subscription service, but there are many textbooks on textile colorants, which give structures of the most industrially useful dyes, e.g. books by H Zollinger or K Hunger. If you have a CAS number, then the usual search engines or ChemSpider may give you a structure, although be wary about accuracy. Failing this, you'll have to treat the colorants as you would other organic compounds in terms of identification, e.g. determine structure through use of spectroscopic techniques, like MS, NMR and FTIR. Commercial dyes are often mixtures and may contain additives (e.g. inorganic salts, dispersant) or contaminants, so some form of purification may be needed first.
NMR and FTIR data are the most effective approaches. Moreover the UV spectra, TLC also may be carried out to verify the speculated structure from NMR and FTIR.