His hands are cold because dude is dead, and has black hands filled with congealed blood. Also worth pointing out: he does not have those sexy blue eyes that the White Walkers get when they're turned.
Now, in terms of the books, the speculation about Benjen being Coldhands was allegedly laid to rest once a Texas A&M library came into possession of an original GoT manuscript in which GRRM's editor asks him point blank if Coldhands is Benjen, and Martin responds "No," which is a pretty hard word to spin as "Yes."
But, in this post-book world, as Kevin Garnett might point out, anything is possible, and so the showrunners decided to tidy things up a bit and combine the two characters. And in case you think I'm just telling a Littlefinger-style truth, co-showrunner Dan Weiss even calls him "Coldhands Benjen" in the after-show. SO THERE.