1st Encounter: Not really much of a boss encounter, she's in a giant mechanical arm. Just stand on the lighted circle and move when the arm is about to smash you. Run up the disabled arm for the prompt. Do it once more and you're done.
2nd Encounter: Same as the first, but with enemies.
Final Encounter: There are 3 things you must pull out from the machine in the middle where Dr. Octopus is (use slow motion for help). Right when you've done that and the force field is down, jump and do a swing kick into her. Now web swing around the level avoiding her as much as you can. She will stop again in the middle. Jump and do another web kick attack just as a giant amount of energy shoots over the floow in a circular pattern. Use the prompt now to hurt her. Keep repeating the steps as needed. The final phase is when she shoots a laser beam at you while enemies attack. Either defeat the enemies or just let them get killed by the laser. Throw the purple spheres they drop at Dr. Octopus until she's out of health.