Go look up vocal fry. It is a speech pattern in which people- usually girls- talk with a tighter and less breathy sound, causing words to sound gravelly and forced.
Not sure how this relates to the personality of John Hamm, actor who plays Don Draper, but typical alpha types (see Ari Gold) usually talk louder and more forcefully, taking huge gulps of air similar to their dominant and forceful personas as if each breath is a war between man and nature.
When cast in this light, Don Draper seems kind of vocally creaky and insecure. At any rate, it seemed unique enough a circumstance to point out the underlying physiological phenomenon.