This is where it becomes very tricky. I had run into this once and you have to hope that the touch screen is still working on the broken LCD.
1. I used an MHL to HDMI adapter (You will need a HDMI monitor and the adaptor to be plugged into a charger for AC power for this) and attempted to use the touch screen to get to bluetooth settings.
1b. I then also disabled screen lock to make things easier when accessing data from the device.
2. Connected a bluetooth mouse just to make things easier for me as the touch screen was damaged in my case.
3. I enabled developer settings (Tapping on build number 7 times which is located in settings > about phone (here or >) > software) and enabled USB debugging.
4. I then used a program called Samsung Smart Switch with the device connected and did a full backup of the phone.
5. Now you can easily restore this backup onto another samsung phone (at least of the same android version or newer).
Not sure if a backup from a newer version of the phone will work on older android operating system.