Is any melachah involved with ripping of dead skin on Shabbat? Is there a concern that one might cause him/herself to bleed?
It is a tolada (corollary) of gozez (shearing). See, e.g., DailyHalacha.com:
The Gemara addresses the related issue of cuticles – pieces of skin around the nail that become detached. The rule established in this regard is that if the piece of skin is mostly detached, and it is “facing upward,” then one may remove it on Shabbat. The Rishonim (Medieval Halachic authorities) disagree in explaining precisely what this means, and therefore the Shulhan Aruch rules that one should not remove cuticles under any circumstances on Shabbat. Since the conditions for allowing the removal of cuticles is subject to debate, we must be stringent and avoid removing cuticles under all circumstances, even if the skin is mostly detached and one experiences discomfort. However, Rabbi Moshe Halevi (Israel, 1961-2001), in his Menuhat Ahaba (vol. 12, Siman 11), writes that in such a case, where the skin is mostly detached and one experiences discomfort, he may ask a gentile to remove it for him.
(Cuticles are dead skin.)
See also here:
Like hair and nails, skin is a part of the body that grows (or more precisely, regenerates). So the principle behind Gozez applies here, and cutting, biting off, or removing pieces of dead skin is prohibited.
See also referenced Shulchan Aruch Orach Chaim 340:2; Biur Halacha s.v. Yabelet; and Rabbi Dovid Ribiat's "39 Melochos" p. 681.
Shemirat Shabbath Kehilchata 35 (32) forbids removing warts or dry skin. Small pieces of skin which are still connected cannot be removed either.