(As of 9 Dec 2013, there is no detailed information about how Headshot Streaks work, but the general understanding of most players is as follows:) You must get 5 headshot-kills in a row without missing, hitting the body, or using a melee attack in-between. In other words, those 5 headshot-kills must not be interrupted by misses or body hits or melee fighting.
Other than that, again as of 9 Dec 2013, it is unknown whether the streak would be broken if you switched to another gun, or whether the 5 headshot-kills must be made within a certain amount of time (i.e. a specific number of seconds), or whether you must remain scoped or aiming for the streak to count. The community is encouraged to experiment and report on findings; and more importantly, approach Undead Labs and ask for the specifics about the mechanics of Headshot Streaks in the game, especially because they are linked to one of the challenges of the Breakdown DLC, the challenge knows as The Sniper, and it unlocks one of the Heroes. Currently, there are several reports about a possible bug linked to Headshot Streaks, as a number of players state that they get 5 headshot-kills in a row with no interruptions, yet the game fails to recognize this, and they receive no credit or acknowledgement of their streaks.
Gives +20 Shooting exp. Does not provide any gun-specific exp (Rifle_Specialization or Shotgun_Specialization). You receive +10 exp for killing any zombie with any gun, so you basically get bonus equal to 2 zombie kills for killing 5 zombies in row. Source: expertise.xml.