LCD Screen is broken, phone is dead & need to access photos?

The phone needs to be unlocked and trusted to the computer for it to allow you to access data on the phone.

With a screen showing no display you won’t be able to do that as you can’t unlock the phone and press the trust button. You’ll have to get the screen replaced first to get data off the phone.

You could get a reputable repair shop to temporarily plug in a screen for the 5C and make an iTunes backup but that isn’t really the best way in my opinion as iTunes backups are usually used to restore on another iPhone (Also it’s hard to transfer iTunes backups from one computer to another) and that overwrites existing data on that iPhone if I remember correctly so better to just get the screen replaced.