This is a very interesting question to consider, as Hamlet does not actually try to use the Bible to justify his own sin at any point in the text. He seems to refer more to a much older, more pagan notion of right and wrong to justify his murder. This more primeval moral compass is based on revenge and violence for violence, and is something that Hamlet finds impossible to ignore, even though he does his best to procrastinate as much as possible and delay killing Claudius. He is well aware that he risks damnation by pursuing revenge, but he feels that the responsibility he has as the son of the old King Hamlet means he has no choice but to pursue revenge. Note, for example, what he says in Act I scene 5 after seeing the Ghost:
The time is out of joint. O cursed spite
That ever I was born to set it right!
Hamlet is well aware that he is placed in an unenviable position and he has very few prospects. Not only does what the Ghost has just told him mean his life in the present is ruined as he has to sacrifice everything, even love, to pursue revenge, but he also faces a very uncertain afterlife, as if he remains true to this older sense of right and wrong he risks damnation with the Christian God, but if he ignores the call of the Ghost he risks being judged by his failure to act in the afterlife. It is no wonder he is such a tormented and troubled figure, as the above quote suggests, as he seems to be trapped between two systems of religious thought and he has no escape: he must please one and reject the other. Therefore Hamlet does not use the Bible to justify his actions in this play.