Reading data from an iPhone 5
Define "heavily damaged". If the screen is crushed just replace it with a new screen and digitizer, swap out the battery, and it may well power up perfectly fine, you can enter in the pass code and go from there. Any cell phone repair shop can do that for less than $100 in a few hours. 

You don't need the charging circuit, the microphone, the camera or even the antenna to be working to view data. You might need the lightning or wifi working to get data off of the device. 

You can probably do it yourself, though the tools might cost more that just paying for the service. They might even have a spare case around to make it look better. 

If the board itself has suffered damage, such as shearing off the BGA chips, it will be considerably more difficult and probably hard to justify unless it contains bitcoins or something like that. The main board only takes up a part of the inside volume and looks like this (from here):