Looking at the evidence we are presented with in the play, I would argue that Hamlet's tragic flaw was definitely not wanting the public to understand that Claudius was a villain. Instead, his tragic flaw was his own procrastination, which is first of all shown through his desire to make absolutely sure that what the Ghost told him was true, but then is exhibited through his delay in taking his revenge that he desires so much. Note how this is shown at the end of Act II scene 2, where Hamlet, in one of his soliloquies, discusses the potential pitfalls he faces if he acts on what the Ghost has told him only to find that it wasn't true. Finally, Hamlet ends with the following lines:
I’ll have grounds
More relative than this. The play’s the thing
Wherein I’ll catch the conscience of the king.
Hamlet here expresses his need to have "more relative" or more convincing evidence of the guilt of Claudius before acting on the Ghost's words, and thus designs the famous "play within a play" in order to set Claudius a test and to discern his guilt or otherwise. However, in Act III scene 3, after he has evidence of the guilt of Claudius, he refrains from killing him because Claudius is praying, and Hamlet believes if he kills him when he is praying, Claudius will go straight to hell. In Act IV scene 4, however, he clearly rebukes himself for procrastinating and not taking up the opportunity of killing Claudius when he had the chance. Note how he compares himself unfavourably to the soldiers he sees who are fighting for nothing more than a bit of land:
How stand I, then,
That have a father killed, a mother stained,
Excitements of my reason and my blood,
And let all sleep?
Hamlet feels that these soldiers, who are so willing to give up their lives for such little cause, stand as a challenge to his own procrastination and inaction, even though he has so much more cause to act and gain his revenge. Hamlet never really acts to convince the public of the guilt of Claudius. He only acts to find out whether Claudius did it or not for his own benefit. In addition, even after he has this proof, he refrains from acting. Procrastination is definitely his tragic flaw.