I've enjoyed playing Galio for quite some time now, and I've thought him to be a great pick against double AP comps for a while. However, after patch 5.6. I'm not the only one who sees his viability. Reducing the cooldown of his Righteous Gust at higher ranks provides him with more utility late game, and helps keep him relevant the entire game. Early, he's hard to gank with his Righteous Gust as an escape and Bulwark to reduce the damage he takes. Mid, he has a good combination of sustained damage, tankiness, and utility. After the buffs, his late game is also pretty good now, with still decent damage and tankiness, but great utility.

Pros:
- great against heavy AP comps
- strong team fight ultimate + utility
- good survivability in lane + easy to gank for
- good damage/tankiness

Cons:
- weak against heavy AD comps
- ultimate easily canceled
- mana hungry early
- only 1 damage non-ultimate ability

Take a standard AP rune page. I'd recommend taking armor instead of magic resist, simply because you will build enough magic resist anyways. Take AP quints + blues for better early game poke, and magic pen reds for the same reason.

Galio is a Mage/Tank hybrid, so your masteries should reflect this. Cooldown reduction offensive masteries are great, so are AP, but you should go 9/21/0 for more tankiness, however utility masteries can also be used. Masteries aren't static on Galio, so you can change them to how you like them. 21/9/0 also works, as does 0/21/9 or 21/0/9, depending on how you choose to play Galio.

You want to have a combination of AP items and tank items to maximize your passive, and get a good combination of survivability and damage. A Spirit Visage on Galio is worth almost 5k gold, and his passive increases the gold efficiency of a lot of items, so take that into account for your build. CDR is very useful, so rushing Athene's or Morellonomicon is highly recommended. You don't want to die quickly in fights, so after you have some early game damage, you should build tanky, and then more damage at the end of your build, so your DPS doesn't fall off.

Max Resolute Smitefirst for maximum poke in lane. This may cause some mana issues, but if you back once around level 5, you should be fine. Bulwark should be maxed 2nd in most cases to give sustain, tower diving potential, and to help your carry and yourself in team fights, but Righteous Gust can also be maxed 2nd if you are team fighting a lot and want the utility it provides, as it can be used to engage or disengage. Righteous Gust can also be maxed 2nd against siege comps for extra wave clear. And, of course, upgrade Idol of Durand at 6, 11, and 16.

: Flash is a MUST have on Galio. A Flash ult is very hard to escape, and it is hard to lane a good ult without flash. Flash is also good for tower diving, escaping, or flashing in to land the final Resolute Smite to secure a kill.
: Teleport is a good summoner spell on most top laners, Galio included. It will allow you to recall early and get back to lane quickly, and can be used to teleport in to impact team fights.
: Ignite can be taken as an alternative to Teleport, although I wouldn't recommend this. If you are against someone who you have high kill pressure on in the early game, you can take ignite, but this is probably the only circumstance in which Ignite in place of Teleport.

Galio is decent in solo queue, but he really shines with team coordination. Even if you're just playing with 1 friend, having that friend camp top with you works, as Galio snowballs surprisingly well, especially against AP top laners. Once he has a first kill and 20% cooldown reduction from the Athene's, he can win most trades, and can even solo kill some AP top laners like Rumble.

Galio is a good situational pick - not someone who should be first picked, but can be a great counter pick against mixed damage or heavy AP comps. He's relatively easy to play, but takes some practice to use his ultimate well and not use it too soon or late. With a good laning phase and midgame, and a decent late game, he doesn't have any points in the game where he's particularly weak and can easily be attacked.