The chemical reactions and pathways resulting in the formation of a ferrichrome. Ferrichromes are any of a group of growth-promoting Fe(III) chelates formed by various genera of microfungi. They are homodetic cyclic hexapeptides made up of a tripeptide of glycine (or other small neutral amino acids) and a tripeptide of an N'acyl-N4-hydroxy-L-ornithine.