If you can't turn the phone on and don't want to risk a battery expolsion, it might be worth going to an independent repairer to see if they can do anything. If you can get power into the phone, connecting to a PC by cable as an external drive is the easiet method. Look for a folder called Pictures, Camera or DCIM.
In future, always make sure you have important stuff backed up. You can set Dropbox to sync photos automatically (limiting it to wifi only if you prefer). Google offers the same facility. At present it's in Google+ and a bit clunky, but it's being moved to be part of Google Drive soon, which makes a lot more sense. You can also put an external sd card in your phone and set that as the default storage for the camera. If the phone fails, you just move the card to another one. Cards can fail too, though, so still set up some kind of cloud sync. :smileywink:
