In Act 1, Scene 1, there is nothing that speaks directly to Hamlet's lack of trust in himself. The scene mostly features the ghost (of Old Hamlet) and how Bernardo, Horatio, and Marcellus react to the ghost.
In the second scene, Claudius and Gertrude question Hamlet about how much he is wallowing in grief for his deceased father. When Claudius asks Hamlet why the clouds hang on him, Hamlet says that he is actually "too much i' the sun." This is a pun on sun and "son." This could be interpreted as Hamlet saying that he is dwelling (too much) on being his father's "son." In other words, he is simply remembering his loyalty to his deceased father; he is dwelling on his role as his father' "son." But Harold Bloom suggests that Hamlet may actually be in doubt that he is his father's son. How long had this affair with Claudius and Gertrude gone on? This is left undecided. Bloom suggests that maybe Hamlet doesn't know whether Old Hamlet or Claudius is his real father. Hamlet, therefore, does not trust his instinct (that Old Hamlet was his father) nor does he trust Claudius and Gertrude.
Horatio tells Hamlet that he once saw Old Hamlet (when he was alive). Horatio adds that he was a "goodly king." Hamlet agrees and replies, "I shall not look upon his like again." Shakespeare is playing here because Hamlet will literally look upon his father again in ghost form. But when Hamlet says he will not look upon a man as impressive as his father, this could be evidence of Hamlet's criticism of himself. In other words, he does not trust that he will become as great a man/king as his father had been. In short, he does not trust himself to become such a great man.