The dragonborn are not a race or a species; no one is "born" a dragonborn. Rather, being dragonborn is a condition, something akin to a disease--albeit a malady of a spiritual nature.
Dragons are manifestations of human sin. Acute greed, the kind that arises out of a gold rush in a mining community, for example, might coalesce into a dragon who cares only for hoarding gold and gems. Monstrous lusts can bring into being a serpent with an insatiable desire to spawn abominations.
Sin shows itself on the face. A marred countenance bespeaks horrid deeds and thoughts, The exterior betrays the interior. Some sinners are so consumed by their misdeeds that their bodies begin to evidence the foulness of their souls. These people become reptilian, squamous, draconic; their bodies begin to grow scales; their teeth elongate into fangs and their fingers curl into claws; some sprout tails or horns. These are the dragonborn.
Here are two notable dragonborn:
Sir Edward Thorpwaite is a minor noble. As a man of means, any of the world's pleasures could be his for the taking but he eschews them all save one: he dines prodigiously and extravagantly. His obsession with consuming rare and costly delicacies has led him to cheat others, betray friends, and steal from those who manage to acquire precious victuals that he has yet to taste. Some say he has even tasted of the flesh of his fellow man. Over the years his sins have resulted in a body that has become a bloated approximation of dragon and man; while fearsome, his razor-sharp teeth seem perfectly formed for sheering through exquisite beefsteaks or munching on petite bon-bons alike. His gut, a scaled sack of flesh that hangs from his torso, always rumbles with the thunder of an approaching storm.
Born to a gravedigger and a washer-woman in some low slum, Karlin Stroud always envied the rich toffs he saw strolling about pursuing the sporting life. Though he was not born to wealth or privilege, he wanted to live as they lived: he wanted a perfumed life of status, leisure, and finery. Adverse to honest labor, Stroud became a sneak-thief and second-story man. His agility and keen eyes allowed him a measure of success; he put his ill-found loot to work by purchasing fripperies and renting a home in a fashionable district. He became a dandy, but dandyisme requires money, and so Stroud continues to operate as a burglar and pick-pocket when not performing as a peacock for the public eye. Unfortunately, it is never enough; he wants more status, more jewels and brocade, and still burns with jealousy when he sees a gentlemen flashier than he is. As a result of this profound envy, Stroud's skin has changed color and he is beginning to take on the scaled appearance that is the tell-tale sign of becoming dragonborn.