This is old fashioned English, so it can be confusing even for modern English speakers. I will rephrase it in a simple and modern way.
"I am certain that old Higgenbotham was hanged and that he is dead. It is not possible for me to believe that he is alive unless I see him alive with my own eyes and unless I hear him say he is alive with his own mouth. That is the only thing that will convince me! You can try to convince me, but if I believe he is alive without seeing this evidence, well, then you can hang ME as a punishment for being so stupid."
So the speaker is saying that he firmly believes something, and he will need very clear evidence if he is going to change his mind. He is so confident in what he believes that he bets his own life on it! He is saying "if I believe he is UNhanged without evidence then you can hang ME".
You can see that I rearranged some things in my sentence. In your sentence the speaker bets his life at the very beginning...
"May I be hanged myself IF..."
If what? He explains in next part... "if I'LL believe old Higginbotham is unhanged".
"I'll" is a contraction of "I will". By saying "if I will believe" he is speaking about the future. For example, "I don't know if I will go to the store tomorrow". So when the speaker says "if I'll believe" he is saying "you can try to convince me and at some time in the future I may believe you".