Why won't my iPhone 5 push calendar information to iCloud?

First check that all your settings are correct, that calendar syncing is checked on all devices (system preferences > iCloud on a mac and settings > iCloud on a iPhone, iPad or iPod).

Make sure the calendars you are using are in your 'iCloud' account and not an 'On My Mac', 'On My Phone' or other non iCloud account (you can do this by clicking/tapping the calendar button in the top left corner of the application ), non iCloud calendars will not sync.

If you are sure that everything is set up correctly and your calendars are in the iCloud account, you might try unchecking calendar syncing in the iCloud settings, restarting your device and then re-enabling calendar syncing settings.