The aggregation, arrangement and bonding together of a set of components to form a melanosome, a tissue-specific, membrane-bounded cytoplasmic organelle within which melanin pigments are synthesized and stored. [ GOC:PARL http://www.ncbi.nlm.nih.gov/pubmed/22511774 GOC:bf http://purl.obolibrary.org/obo/go/references/0000079 GOC:TermGenie ]