Questions: is it the display that's dead, the touchscreen, or both?
If just the display then you can use a MHL cable (check compatible with Samsung devices before buying) to display on a TV or monitor. Then you can control it using the touch layer.
If just the digitiser, connect a mouse or keyboard using a USB OTG cable and control the phone using that.
If both then it gets awkward: the problem is that both MHL and OTG need to use the USB port, so you can either see what you want to do (what menu you are in, what icons are where) or you can interact with the device, but not both at once! I actually did recover data from a device in this state last year: the trick is to get past the screen lock and pair it with a Bluetooth mouse or keyboard. Then you can mirror the display to a TV and control the phone using Bluetooth. But getting to that point requires patience!
Once you've done that then you upload your files to cloud storage, or install something like WiFi File Explorer or Airdroid that will let you copy stuff off to a computer wirelessly - since the S6 has no sd slot and we're still using the USB for the display! Or enable USB debugging and then use the USB port and adb to copy stuff as Luna suggests.
Anyway, I'd not say that it's easy (not if both are broken), but it is possible as long as you have the patience...