First off, you're overthinking it. Press the barrel of a decent caliber handgun flat to the side of your head basically anywhere between the temple and the back of the ear, pull the trigger, and you're gone. If you're being extra careful, make sure you've got 5 minutes to bleed out before the paramedics arrive, but even that almost never makes a difference. People who survive self-inflicted gunshot wounds to the head tend to do so because they miss (or mostly miss) the brain, for example by shooting through the bottom of the chin, or putting the gun in their mouth or the like. And even a fair number of those are fatal. It's a highly lethal method, and you're probably better off to make sure you have a decent sized bullet and a large enough bullet path through brain material than you are trying to find the sweet spot to take out the brainstem.
Having said that, the basic issue with a suicidal gunshot wound to the head is that you have a number of tradeoffs to make. In general, though, a medium or large caliber round that crosses the sagittal plane (side to side) is almost always lethal, and any bullet that hits the brainstem is absolutely lethal. The brainstem is harder to hit, but as instant a death as you can get. Any other gunshot to the head actually kills by exsanguination (blood loss), though fortunately, for how much blood the brain uses, it takes a few minutes at most.
So, the easier shot (side-to-side, crossing the sagittal plane) is a slower death, the harder shot is instant. If your bullet trajectory is a bit off crossing the sagittal plane, it won't matter, if it's a bit of going for the brainstem, you might end up a quadriplegic, or as one of those heartbreaking case studies where someone had to shoot himself in the head 4 times to die. Trade off number one - brainstem or sagittal plane.
Ideally, you want a large caliber, high velocity round, which translates to a decent caliber rifle, or a shotgun slug. The problem with that is that the ideal round comes from a less than ideal gun - it's about impossible to make a side-to-side shot with a long gun, and going for the brainstem with a long gun means taking the through the mouth approach, which is harder than it sounds and can result in pretty horrific injuries in the event of a miss. (Intense burns on the inside of the mouth, facial disfigurement, permanent damage to the throat, full paralysis, etc.) Alternately, a handgun has a less than ideal round, but a much easier shot at either at the brainstem or crossing the sagittal plane. Trade off number two - better round, or gun that's easier to aim.
Then, there's the issue of what caliber round to use. If you've ever shot before, you know that generally speaking, the higher the caliber, the more recoil there is. Even small children can control the recoil on a .22 rifle, but it takes experience to shoot effectively with something at the other end of the range, like a .45 or a 50 cal. When it comes to a self inflicted gunshot wound to the head, that matters more than it would seem like it should - even experienced shooters tend to flinch a little when they're shooting themselves, and it's not possible to use the proper shooting stance and grip when aiming at one's head, so recoil matters. Trade off number three - higher caliber for more lethality, or lower caliber for more control.
Not my method, but if it were, I'd take a handgun and go side-to-side, rather than aiming at the brainstem. I've read too many horror stories about people who just ended up horribly injured after missing the brainstem.
