Gragas is a very strong champion and in my opinion one of the most fun to play. I play him just like his lore says. Get in, do large amounts of damage, and then get out. One of Gragases advantages is his laning power. His passive gains his HP back as his w helps keep your manna up.

Gragas has very strong base ability damage throughout the hole game as well as having strong AP ratios. Another one of Gragas's strong points is his ability to farm. By level 4 the combination of q and then an e into the ranged creeps will result in a huge gold difference between you and the oposing team.

By this point you should have boots and a Death Cap and starting on your void staff. Against most champions you should be able to poke them to half health and finish them with an r,q combo. When playing Gragas you'll have a range advantage with your q. You can use this to push back your enemies back and keep them from farming. Which will contribute to a large level and item lead in the late game.

By this point you want to finish your build and wait for a time to strike. If you are having trouble finishing your build try using your e (body Slam) to move between lanes and get those minion farms. Also at this point its all on you. Continue poking the enemy during those turret stand offs, have your team wait while you widle down the enemy's health, once their health is around 50%-75% and use your ult. try to split apart their team.

When buying items buy the higher AP items first. If you cant buy a Large Rod right away buy another dorans ring.
Dive through walls to gank/runaway.
BE GOOD AT SKILL SHOTS. Do this by leading your targets and predicting movements.