Flash: Obviously. You can't live with it and you won't live without it.
Teleport: Get to your team after you solo baron, or teleport to Baron as your team is dieing without you

Zhonyas first. Your strongest weapon is your opponent's rage at dealing with your turrets. They go in for the kill and you just pop Zhonyas and they die to your HG38's as they confusedly wait for you to pop out of your immunity.
Next is your key item to kill Baron, Torment. Percent health is really what destroys him.
After that it's items to ramp up your AP which helps with all your magic penetration. And if you have full build with Rabbadon's DeathCap you'll have AP out the ying yang.

Last is your Omnivamp item, because your turrets will die to Baron even with a critical upgrade. Practice on soloing dragon early and it just gets easier from there.
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As for soloing Baron Nashor: Wait until the enemy team is distracted so they don't just walk up and kill you. Make sure you have Torment and omnivamp and are at least 16. that is how you stay alive and maintain your health. I recommend getting close to full build, 5/6 items is a necessity. You can skip Void Staff if you think you can manage. Testing is important in the proper environment.

Place your upgrade turret first, followed by your next two. Your omni vamp will keep you alive and high AP torment build will shred his health. Your turrets will still die but its important to take some damage for them, but hey, thats what the omnivamp is for. Just as your team retreats to your towers after abandoning a team fight, you'll all get Baron buff and your enemy team will wonder why they were laughing at your selection of champ.
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Furthermore, you can even jungle Heimer if you're brave enough, but I would not really recommend it. So why mention it? Because you can solo dragon at 6 without any real issues, and smite is perfect for making sure Baron isn't stolen.

Turrets are your best friend, and secondly, storm grenade stuns. When not fighting Baron ( which is your god-Forsaken job), you should be using upgrade in team fights on H-28G Evolution turrets, except when you can land a really sick Micro-rocket or Storm Grenade R, whichever will land the most damage. The easier one to land is your Grenade, which has more range and can be executed quickly.

You still have to beat Mid, which means pressuring them with your turrets. Your best bet, as always, is to get an advantage early, which is why I go with scorch and arcane comet. Manaflow band is to compensate for your mana issues, and Absolute Focus is because they will not want to approach your turrets, giving you a health advantage ( dont ask me how that works ( Ill tell you anyway, it's because you use their fear to leverage damage against them)). The rest is just more AP and health, which is great against all kinds of damage.