YOU'RE A G.O.D.D.A.M.N. DRAGON.
Patch 6.19 nerfed the hell out of Astral Flight, so early game will be much tougher, but Aurelion Sol still very much melts everything in sight by lategame.

Pros:

[+] Constant damage output because of Cosmic Creator
[+] Great teamfighter because of sustained damage from Astral Flight and crowd control from Breath of Light
[+] Excellent at ganking / roaming from Singularity, which doesn't have to stack anymore since Patch 6.19

Cons:

[-] Not much defense against assassins
[-] You're a huge target for skillshots (but your tail is untargetable)
[-] Awful against teams full of hard CC since they will disable your stars

The upgrade path for our star dragon should almost always be R > W > E > Q.

MAX YOUR W FIRST!

I can't emphasize this enough. Three stars orbit you for a reason. Use them since they are your main source of damage. Every level of Astral Flight increases the passive base damage of your stars (only 5 instead of 10 after Patch 6.19 ://), and activating the ability gives you +50% MORE damage. You can literally go AFK and get a kill as long as someone runs into your orbit. GG EZ.

Get Your Roam On

Singularity allows you to fly almost globally (from mid lane, your radius covers the entire map at max rank), which makes you a natural roamer. Fly in for easy ganks to either side lane, and your jungler can basically camp the other half of the map. This is why I recommend getting this ability to at least level 3 before you start leveling Breath of Light. The counterplay is insane.

The Ultimate

Upgrade your Falling Star whenever you can. It actually doesn't do much damage because its main bonuses are the slow and knockback, so any extra upgrade will ensure that you secure the kill of anyone that tries to run.

The Star Stun

Max Breath of Light last. Neither the stun duration increase nor the damage increase is very significant between levels, so this ability is a one-point wonder, just like Katarina's Shunpo.

Aurelion Sol is relatively versatile with regards to spells, unlike someone such as Katarina who pretty much requires Flash and Ignite, though the following two will offer you the best utility. Possible alternatives are listed below, but they might not be as useful to the star dragon.

Barrier is your best friend against those jumpy burst assassins that are all-too-common in the mid lane. This is especially useful since your Breath of Light has a small delay before it can be exploded. Lee Sin shows up to gank and jumps on you? Barrier. Leona pulls herself toward you? Barrier. Yasuo actually managed to tornado you and is trying to pull of his ultimate Last Breath? Barrier.

Exhaust is a very nice addition to your Crowd Control Hell. Not only is it useful during teamfights, but it also keeps enemies within your orbit, so you can immediately proceed to stun them and harass with Astral Flight and maybe follow up with Falling Star for a free kill. It's especially useful pre- Rylai's Crystal Scepter so no one escapes you with 2 HP.

Alternatives

People will argue that everyone and their grandma should use Flash 100% of the time. It offers clutch escapes and useful gap closers, they'll say. HOWEVER, since Sol already has so much mobility for both chasing and escaping with his momentum-based movement speed increases, Singularity, and Breath of Light for that +10% movement speed, Flash just isn't completely necessary.

Ghost is useful for champions who rely on chasing to secure the kill. For example, Master Yi's Highlander & Ghost combo grants him about 700 movement speed. HOWEVER, as with Flash, Sol already has enough speed built into his kit.

Cleanse is an alternative to Barrier if the enemy team is packed with HARD CROWD CONTROL, like Blitzcrank's Power Fist, Sion's Decimating Smash, and Malphite's Unstoppable Force, which disable your stars.

Ignite can be taken if you're feeling overly aggressive and want to kill secure, especially in the early game, which was hella nerfed by Patch 6.19. Unless you rush a full AP item instead of something tankier like Rod of Ages, your early damage is low and the increased mana costs, especially for Astral Flight are insane.

Don't Take

Sol naturally scales as an offtank so you personally won't have trouble with HP, and it's likely that the ADC and/or support already have Heal anyway, so it's a waste of a slot.

Teleport can very much increase map pressure, but if Singularity is decently upgraded, you can fly almost globally and be present to help your team in almost roughly the same 4 seconds that it takes to teleport.

• Quintessence of Ability Power and Mark of Magic Penetration for that beautiful AP damage
• Seal of Health to stay alive early-game, and Seal of Scaling Health for later on, so you can sustain your boop damage in teamfights for longer.
• Glyph of Mana Regeneration because your Astral Flight eats mana every second. The longer you can keep it active, the more damage output you have.
• Glyph of Magic Resist, but the defensive items will offer that anyway.

Alternatives

• Glyph of Ability Power if you're lusting after pure power.

You've probably heard this for many other champions, but Thunderlord's Decree is absolutely the best mastery to have on this guy. All it takes is the Breath of Light stun > an easy comet boop > auto-attack combo, or even three consecutive Astral Flight boops to activate it.

As for the others, Feast (2nd tier Ferocity) can be swapped with Expose Weakness if you're planning on roaming a lot and/or playing more of a support role. Also Merciless (3rd tier Cunning) can be swapped with Meditation , but after early-game you won't be running into mana problems.

Starting

Doran's Ring, Health Potion, and Stealth Ward are the standard AP Mage starting items, perfect for that initial damage. You don't have to fly across the map just yet, so boots can wait.

Start Rushing Core Items

Ideally save up 1100 gold for Catalyst of Aeons before recalling. It'll compensate for your lack of health and especially mana early-game. Both Rod of Ages and Everfrost offer roughly the same health, mana, and AP stats, so your decision depends on their effects. Pick the former if you want to become more tanky over time due to the passive, or the latter if you want a slowing active effect. Both items restore mana and health, and both synergize nicely with Archangel's Staff, which awards AP based on mana.

Boots

Boots of Swiftness are the best because they allow you to make godly escapes, along with your movement speed stacks.

Mercury's Treads if their team is packed with hard CC.

Sorcerer's Shoes are not as worthwhile as the other two, but are still an option if you're looking to go full mage.

Rylai's Crystal Scepter

At the 20 minute mark (or closer to 15 if you're a lot better at last-hitting minions than I am) your life will change. This is when you'll finally be able to afford Rylai's Crystal Scepter, a.k.a. the

most important item you can build on Aurelion Sol

. The bonus health and AP is a nice boost for your teamfight performance, but its defining feature is the passive. Every single one of your Cosmic Creator boops will slow enemies. Along with your insane mobility, this item will make it nearly impossible for enemy champions to run away from you, as well as try to chase you, without Flashing or dashing.

Late-Game Defense

Abyssal Scepter - If you are getting stunned (from Ahri's Charm, Veigar's Event Horizon, Xerath's Shocking Orb, Ryze's point-and-click Rune Prison, etc.) and then absolutely melted, this is a perfect choice to increase your chance of survival while offering a decent amount of AP as well.

Zhonya's Hourglass - Either for teams full of AD--like Zed, Master Yi, and/or a fed ADC--or even just for the active Stasis. Allows you to fly into the middle of their team, stun, and then shield yourself for a few seconds while your team eats them for breakfast.

Banshee's Veil - If you are getting melted by the other mid laner (like by Veigar and his stupid point-and-click ult), and want to become a straight up tank, buy this. It's like a less insane version of Warmog's Armor.

Guardian Angel - If you're not getting particularly melted by either an AP assassin or mage, or the enemy ADC, but still want to build more tanky, this item will offer you the best of both worlds. Plus the revival passive can completely save teamfights.

Late-Game Offense

Liandry's Torment - Probably the best / safest choice out of the rest. Just read the description:
+80 Ability Power (you will still melt)
+300 Health (even a better choice than some of the defensive items... screw their True Damage)
+15 Magic Penetration (not as good as Void Staff but who cares)
+Sustained fire damage, which IS DOUBLED WHEN THEIR MOVEMENT SPEED IS IMPAIRED. When paired with the slow from Rylai's Crystal Scepter, no one will escape you alive.

Luden's Echo - Cheaper than the Deathcap, offers almost as much AP, as well as some extra Movement Speed, and the on-hit effect means that your boops will occasionally echo damage to multiple targets for some area of effect fun.

Void Staff - If the enemy team is a bunch of weak b.i.t.c.h.e.s who think they can counter you with a couple points of Magic Resist, they have never been more wrong in their life. Grab this item for the Magic Penetration and nothing will stop you.

Rabadon's Deathcap - If you just don't give a s.h.it. anymore and simply want to MELT EVERYONE IN YOUR PATH, this is the perfect endgame mage item. +120 AP means three boops and the ADC is dead. Everyone is dead. You will carry the game.

Archangel's Staff - More mana sustain to keep your Astral Flight going longer, and it also gives more AP based on total mana, so you basically become Ryze.