This would be pretty hard to do, actually. It’s not so hard to tear out hair and probably scratching might be possible (and probably that’s what Hamlet actually means here) but to tear one’s self, like one would tear a piece of paper or even a piece of clothing would be.
Trying to tear one’s self might be a lot like that silly alt-right fella trying to tear up a protest sign and just trying over and over as hard as he could and never managing it. A study in futility and a delightful episode of schadenfreude.