Do you doubt that? Do you think it’s out of the realm of possibility that a man who personally poisoned his brother would hesitate to have his enemy killed at a remove?
It may have even been more surprising if Claudius had NOT attempted this sort of villainy. It’s not like Hamlet has some kind of magical 6th sense. He knows what his uncle is capable of. He knows his methods. Of course that letter to the king is a death sentence for him. Horatio should maybe not be surprised.