Securely wiping a phone with a broken screen
If you did a factory reset, you should be able to connect it to the computer and it should allow you to delete files from the internal storage. MTP is limiting though in the fact that you can't use a secure erase program (one that overwrites the file multiple times with random data). This will be quite difficult to do with a broken screen. 

The other option is to use Odin to restore it to stock. Odin overwrites all the partitions including the full /data partition which will wipe the internal storage. Just follow any of the guides on how to return to stock or unbrick the device, like this one for the international S3. If you have one of the LTE or carrier versions you will need an appropriate guide for that version. You can do everything blind in Odin, just follow the directions carefully.