Even in the first two scenes of the play, there are indications of Hamlet being "unreliable." Where might you see such lapses in reason and "fact" occurring?
Throughout the play, Hamlet's passion and emotion seem to get in the way of any reasonable action or point of fact. He often focuses on his own pain, wishing death or suicide upon himself. In a violent fit, he stabs Polonius. He rambles on to Rosencrantz, Guildenstern, Polonius, the King, and the Queen causing everyone, including the reader, to question his sanity. He arranges the murder of those who are supposed to be his closest school friends. And he tortures Ophelia with a passionate 'I love you, I love you not' charade. Because he is so overcome by his emotions throughout the play, and because so many of his actions are based off emotion, I think he becomes unreliable. Even in soliloquies where he is trying to reason through what he should do, his emotions overtake him, and hinder any effective action.
I think we can see some unreliability in his first soliloquy (1.2). First of all, he starts out by wishing that God had not said that "self-slaughter" (suicide) was a sin...so we have a very depressed, suicidal young man - not necessarily a picture of sanity and reliability.
Then as he's talking about his mother's hasty remarriage, he initially says that it was two months between his father's death and his mother's remarriage. But as his speech progresses, he lessens the amount of time between the two events - first it's "but two months dead--nay, not so much, not two." Then "A little month...", then "Within a month...".
The way I, personally, read/hear this soliloquy is that he starts out calm and quiet, but slowly becomes more enraged, more disgusted with what has happened, until he ultimately loses some credibility because he's not thinking through what he says before he says it - he's just reacting to the pain and grief he's feeling inside.