The dye trader will appear soon after you pick up a dye or item that crafts a dye as long as you have suitable housing.
You will need to purchase a Dye Vat in order to craft your own dyes.
His initial inventory includes:
Silver Dye: 1 gold
Dye Vat: 1 gold
A Suspicious Looking Eye: Kill one Eye of Cthulhu
--Upon getting the quest "A Suspicious Look Eye" it says "You have received 'a suspicious looking eye'". But it never appears in your inventory.
--seems that he doesnt accept strange plants.. no idea if its a bug, but he doesnt take them.