Hamlet what were you thinking?

Sometimes I question Hamlet's character. He's so paranoid. I dont know if I would've approached the situation the same way he did. Considering his only shred of evidence is a dream in which his father appeared to him in, he shouldn't be making any rash decisions. If I were him I would've just began with a simple rational plan. Start off by trying to get some tangible evidence, or maybe at least try to hear something suspicious that his uncle might tell someone else. After this, if I found credible information and substantial evidence, I probably would've figured out a way to kill him without having anything put on me. My best option would have to be to poision him in his sleep. No one would know it was me, I would just have to worry about doing it stealthily and carefully.