Actually, its resemblance to deadliness is not all that coincidental. One of the meanings of the word deadline is "A line drawn around a military prison, beyond which a prisoner is liable to be shot down. orig. U.S." The earliest usage the OED lists for the word is 1860, for the meaning "a line that does not move or run," but the next earliest one is the military definition: "1864 in Congressional Records 12 Jan. (1876) 384/1 The ‘dead line’, beyond which the prisoners are not allowed to pass."
If I recall, the "dead line" was well known at Andersonville in Georgia, a Civil War prison camp that housed (I use the term loosely) something like 30,000 Union soldier POW's. If any of them even touched the outer wall, they were instantly shot by guards. There were occasions on which a prisoner, mentally unable or unwilling to survive another day in the place, would commit suicide by purposely touching the wall. Andersonville was an active site, again if I recall, around 1864. (I haven't looked any of this up to verify, so apologies if some of the details are wrong.)
Remembering this usage probably helps one meet the more prosaic deadlines.