Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
This build focuses primarily on maximizing Gragas' strong health regen from ability use and his strong laning presence early game.
Try to lane with a carry of some type and do your best to feed them!
Early game: Try not to blow too much mana on body slam until you've got at least one rank in drunken rage. Gragas' body slam in conjunction with ignite makes for an easy first blood in most cases, but don't overextend too far for a kill. The more aggressive you are early game the more scared the opposing lane members will be. Once you hit level 6, try to catch one of the other team members off guard and body slam -> explosive cask -> body slam -> ignite for an easy kill with a little help from your carry teammate.
After your first kill or once you gather enough gold to purchase Spirit Visage and regular boots, return to base unless the *hopefully* carry in your lane needs your help.
Mid game: Usually by now you can run in at body slam enemy teammates at will. Even if you're dropped to half health you can just body slam / drunken rage a few times to be right back up to full. Begin moving around the map to help with ganks, and don't be afraid to tower dive to get that last 100 health on an enemy.
Item progression depends upon enemy teammates, but for the most part the items I have laid out are a well rounded development. Boots are purely situational, but I usually go with merc's for the slow reduc.
Late game: Once you finish Omen and Force of Nature, no one will be able to kill you unless you stand in 2 turrets worth of damage stunned the entire time. Watch out for enemy team carries if they have a stun of some type, but for the most part, body slam in and harass as much as possible. If you've got the time, grab lizard buff for the slow and the boost to your regular attacks.
The last two items are really up for grabs. If you feel like you're team is doing hard work and you haven't come close to dying, grab a frozen mallet and a rylai's for the stupid amount of slows you'd be able to put up. If the other team's melee carries are wrecking you, grab a sunfire cape or a thornmail, or even both! If you find yourself getting slammed by a Malzahar every time he ults you, grab a quicksilver sash and a warmogs to offset the damage and get out of the null zone.
Anyways, this is my first guide, so much feedback would be appreciated. I've played nearly half of my games played with Gragas, and, although he cannot carry a team to a win on his own, he CAN set up many easy kills for your own team's carry. Don't forget to play aggressive, and the build should work out in your favor!
Thanks for reading :D I'll continue to update this guide and to answer any questions in the comment section.