Gragas although very Tanky has very high potential as a AP carry, and is at the moment my favorite champion for this position. Gragas does best in mid lane or top lane depending on the opponent however in general mid should be your default choice.

getting both pots and finalizing the build gives you 4000hp, 70mr, 130armor, 15mp, 800ap, an invulnerability, and a passive 18% reduction on all damage, which applies before armor... with these stats gragas is easily the best end game mid, once finished your build you will do easily more damage then most other Mids, as well as being a unbelievable tank, and having more team fight utility then any other mid.

Summary
- GRAGAS = GOD, Do a Barrel Roll Gragas, Do a Barrel Roll!

runes are fairly straight forward the AP you gain from them allows you to start mass farming CS very early game and gives you a much stronger harass, if you choose other rune types over AP it will not end up being the end of the world however it will take you much longer in order to start one hitting minion waves with your Barrel Roll, the reason we use Greater Glyph of Scaling Ability Power is because gragas is more of a late game champion anyway, you will not be able to do much before lvl 6 which is when scaling glyphs start giving more AP then static glyphs.

Summary
- Runes are extra free stats, use them!

fairly straight forward, the core of the offensive tree is important, however ability choices in the second tier of support is mostly up to you, i find that movement speed is more handy with chasing and escaping then the worth of a little bit of mana late game but as i said, it's up to you overall.

Summary
- Masteries provide extra free stats, use them!

gragas has 3 attack type abilities 2 of which are also good for escaping or team assistance, his 4th ability is Regen + Tankiness + Dmg. his Barrel Roll is his bread and butter it has a high AP scaling and does large amounts of AoE Dmg. his Body Slam is good for Dmg, chasing, dodging, running, and travel(late game it can be used constantly with a 5 second Cd to allow you to travel at a very fast pace between lanes, allowing you to leap over walls and just in general move faster). his Explosive Cask is a very high Dmg AoE knock-back, it knocks everyone in range of the blast away from the center - it can be used for ganks, escapes, rescues, and in order to separate the enemy team in a team fight allowing your team to gain the upper hand sending several flying away and a few flying towards you creating a 5v2 or other similar scenario. his 4th ability is his Drunken Rage, when gragas drinks he gains mana, attack Dmg and Dmg reduction - you will want to keep this up at all times.

Summary
- Barrel Roll is primary Dmg/Harass/CS
- Body Slam is Escape/Chase/Dmg/Travel>(when you have mana to spare)<
- Drunken Rage is Mana Regen/Hp Regen/Tanky/AD
- Explosive Cask is Dmg/Escape/Support/Gank.

gragas can build many different items in order to create builds that can function better in different scenarios, however this build focuses on becoming both an AP canon and a strong off tank. the build will end up giving you a large amount of AP, and decently high defensive stats.

Athene's Unholy Grail - much needed regen, very high amount of early game magic resist(with the magic resist+your Drunken Rage you take almost no DMG from other mids), CDR and a nice amount of early AP
Ionian Boots of Lucidity - movement speed and enough CDR to max yourself
Rylai's Crystal Scepter - increased HP, a very handy slow and of course more AP
Liandry's Torment - HP, Magic Pen, a DoT and more AP!
Rabadon's Deathcap - MORE AP
Warmog's Armor - seriously increased tank capabilties, giving you enough regen and hp to essentialy start ignoring haras and be very aggresive
Zhonyia's Hourglass -(sell boots)- a very strong use effect that gives immunity to damage, increased armor and a ton more AP

Through the combination of items you have decreased CDS( Ionian Boots of Lucidity+ Athene's Unholy Grail+masteries+blue pot late game) on your spells and every time you land your Barrel Roll you will put a large dot( Liandry's Torment) on the target as well as slowing( Rylai's Crystal Scepter) them making it easier to chase down enemies and strike them with other attacks. this slow can also be used while fleeing, in order to assist allys, or escape yourself. You toss down Barrel Roll so that chasing enemies have to travel over or around it to get by if they try to go over you slow them, both harassing and providing a chance for escape. Warmog's Armor gives you enough of an increase to start off tanking, you will make most bruisers look squishy by comparison even. Zhonya's Hourglass getting this makes you more tanky and increases your ap to 800 with blue pot, as well as giving you an immunity if you get in trouble

Summary
- 40% CDR + 800 AP + Enough mana to sustain yourself + higher then bruiser tanking capability = strongest mid in-game.

for the lane phase you want to maximize gragas' potential at CS, in every game you play you should get top CS by a wide margin because in general if used properly his Barrel Roll will be able to get you a full clear of minion waves with a well aimed toss, you can do this even if your just passing by on your way to team fights allowing you to mass quick CS, as well as keep multiple lanes pushed.

Summary
- CS is good for you, use your Barrel Roll!

during the laning phase you want to be fairly aggressive, if you can aim your Barrel Roll properly you should be able to scare off your opponent after just a couple hits. first harass your opponent to scare them off, you will want to keep using your Barrel Roll in this manner whenever you are able to get a hit HOWEVER - your priority is to make sure you get all your CS and keep the lane pushed; by level 8 you should be able to get almost all your CS by using your Barrel Roll. whenever there is an open interval you will want to start also clearing wraiths, the process for this is fairly quick and easy as soon as you clear your wave of creeps and are waiting for more to arrive you walk backwards to the wall just south(or north) of the wraiths and toss your Barrel Roll; you then Body Slam through the wall to the wraiths and activate your Barrel Roll as you hit them with your Body Slam. after killing the wraiths you Body Slam back over the wall and return to lane.

Summary
- stay only as close as is safe(if you are more fed go closer). use your abilities wisely, if you spend your mana poorly you will go OOM.

blue buff is not necessary for Gragas; HOWEVER - if you have blue buff this allows you to spam your abilities for harass and CS which causes you to heal more Happy Hour allowing you to sustain your lane without ever going back AND allows you use your abilities more often, which is huge for Gragas with the blue buff you should be able to cast your Body Slam on a 4 sec CD early game making ganking you an almost impossible task even for multiple incoming enemies.

Summary
- blue buff, fantastic and very very beneficial(you should ask for the second one onwards every game), but if you cannot get it you should be able to handle yourself still.