Hello. Im Holom and this is my first guide. in this guide I will help you play Gragas with great survivability and even better Damage with Ability power.
I play Gragas as solotop because of his harass skill and great farming spells of Barrel Roll + the Attack Damage from Drunken Rage and of course his Happy hour passive that restores 2% of his health back in 4 seconds every time he casts a spell.

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The runes are pretty simple. Every day caster runes

Greater Mark of Insight : Gives 0.95 magic penatration for Barrel roll + Explosive Cask ( 8,55 Magic Penatration)

Greater Seal of Clarity : Gives 0.065 Mana regen / 5 sec per level and 1.17 at level 18 ( 0.585 Mana regen/sec at level 1 and 10.53 Mana regen/sec per level at level 18 ). You can always switch these to Health or something else because you do have your Drunken Rage that gives you a bit free mana every time you cast it.

Greatet Glyph of Force : 0.17 Ability power per level and 3.06 at level 18 ( 1.58 Ability power at level 1 and 27.54 at level 18 ).

Greater Quintessence of Potency : 4,95 Ability power ( 14.85 Ability power ).

9/0/21 Usual Caster runes. You can pick the extra Experience instead of the Mana and Switch Greed to Increased neutral buff time but this is how I roll.

Your usually best starting with Sapphire Crystal and few HP Potions or Doran's Ring for extra HP + AP and build Rod of Ages as fast as you can. After that Boots and continue with Rylai's Crystal Scepter by this time you should have nice amout of HP and start building towards Rabandon's Deathcap for great AP Boost for the last 2 items you should look what kind of damage your enemy is doing if they're mostly AD = Armor and if theyr Casters = Magic resist.
Always remember to buy Wards when you go back and place them to river so you wont get ganked!

In Keyboard = Q - W - Q - E - Q - R - Q - W - Q - W - R - W - W - E - E - E - E

BARREL ROLL (Q) is your main source of Damage. 85/135/185/235/285 damage 90% AP rotation - 11/10/9/8/7 CD - Slows enemies attack speed with 20%/25%/30%/35%/40%
it has long range ( 1100 ) so you can just shoot it around without problems that enemies would attack you and if you got ennough Ability power you can kill Minion waves instant with this and this is really powerfull spell in Teamfight becouse of the AoE Damage / Slow's the attack speed enemies

DRUKEN RAGE (W) Is your way to gain Mana / AD / Attack reduce. Always Spam with spell when ever you can especially when entering and In Team fights to take less damage!

BODY SLAM (E) This spell does fine AoE damage and Slows the enemy but the the main purpose of this spell is that you can chase or escape with it. You can use it trough Tree's and cliffs.

EXPLOSIVE CASK (R) This is your Ultimate With almost same range of Barrel roll and greater Damage when you hit it in the ground try to hit it front of Enemy champions so they will be knocked to you and your team This is great in Catching the runners / Running away / Starting fights / Interrupting enemy's Ultimates.

There few Summoner spells you can take to Gragas :

IGNITE : Gives you extra damage.

FLASH : Not that great becouse you do have Body Slam that does the same as Flash but has lower Cooldown but its still fine.

CLARITY : You have your Druken Rage that gives you some mana but extra mana is always welcome.

TELEPORT : Gets you faster to your lane.

CLAIRVOYANCE : You rarely see Gragas picking this but its kinda good. Extra map awarness and you dont need to facecheck.