The OED suggests that it comes about by extension from terms such as 'dead asleep' or 'dead tired', where a comparison to death is actually being made.
The first definition of 'dead' as an adverb is given thus;
"In a manner, or to a degree, characteristic of or suggesting death; with extreme inactivity, stillness, etc.; utterly, profoundly, absolutely (as dead asleep, dead calm); to extremity, ‘to death’ (as dead run, dead tired)."
The dictionary then goes onto provide this second definition (based upon the first one):
"Hence more generally: Utterly, entirely, absolutely, quite. Esp. dead broke (see broke adj. 3a), dead certain, dead easy, dead frozen sure, dead level, dead right, dead sure. Now colloquial".