you want to shoot through your mouth for sure, not directly back through your throat/neck, but a little bit up, so you're aiming at the back of the roof of your mouth if that makes sense. aiming through your mouth towards your ears would do it as well. you definitely don't want to shoot straight up in your mouth either, that might not be fatal.
you are aiming for the brain stem, you will be clinically dead instantly the second your brain stem is severed.
any bullet would be sufficient for this. 9mm would be fine as long as you aim correctly, the only point of using a shotgun is that you can't miss.
If you want to be sure you are going to be dead, best to combine the shot with something else that is fairly quick (so you won't be discovered and saved). partial suspension hanging or 60+ foot fall onto cement should do the trick.