The Amazing Spider-Man

To clarify on C480o0s3's solution, Spider-man has normal upgrades and tech upgrades. Normal upgrades unlock as you complete the story missions and you should have most (if not all) by the end of the Oscorp Bio-lab level (where you save the scientists with the second antidote). These are purchased by gaining experience and leveling up to a max level of 20.

Tech upgrades are unlocked as you beat secret labs. Once you beat all four secret labs, you should have every available tech upgrade for purchase. Tech points are obtained by destroying turrets and sentries as well as large scale robotic bosses like the S-02. There are also hidden tech containers inside most levels (which will net you another achievement for picking up), which can help upgrade you as well. As long as you don't avoid killing sentries and are a bit explorative, you should have enough tech points at the end of the game to upgrade to max.

Be forewarned that there are rumors of a possible glitch that doesn't unlock the upgrades at the normal time. If this happens, it is suggested that you replay story missions in an attempt to unlock them, though you'll most likely have to restart the game.

Plain and simple, just get all of Spider-man's normal and tech upgrades. To speed up the process, find collectibles and do side missions throughout New York and the main missions.

UPDATE:

The upgrade that allows you to tack the asylum escapees is glitched. Sometimes it will unlock, other times it doesn't and the only way to fix it is playing the game from the beginning.