How do I gain the right to access my neighbors property to remove a hazardous branch coming from my property?

Call your homeowner's insurance agent, tell him the situation and ask him the proper way and legal way to handle this so that neither you nor your insurance company will be held liable for any damage this tree branch may cause to your neighbor's property. Your neighbor's homeowner's insurance company may be very interested in knowing about this as well. Your neighbor leaves you no choice but to take action to protect yourself. After you have protected yourself, let the tree fall where it may and let your neighbor worry about it.