To start the fight, you need to walk below a certain line. It's possible to get 3 bombs in if he shoots fire, two if he drops spiders, and only one if he opens the pits. He dies in 5 bombs.
In the Japanese version, if you throw bombs near the center of the room before he appears, you can kill him without actually starting the fight.
Dodongo has two possible attacks: fire and charge. If he tries to shoot fire, you throw a bomb at him and throw him on the spikes. His pattern is always fire, fire, charge, fire, charge, fire.