Does the device support USB OTG and does it support display mirroring of any sort (e.g. MHL)?
If it supports OTG then you can use a USB keyboard or mouse to unlock it and enable whatever you need (set file transfer as the default usb connection, enable usb debugging, run a WiFi file transfer app that you push to it from the play store website, whatever). If it supports display mirroring you can use that to see what you are doing. The tricky bit is if you need to use the same USB port for both! I've managed to recover data from a phone with a dead screen by swapping between a USB keyboard and an MHL connection (type some keys, swap to seeing where you are, work out what to type next, swaps again, etc) but it takes patience. Enabling bluetooth and pairing with a BT keyboard (not simple) made things a lot easier. But it depends what the device supports (and also whether it's just the display not working or whether the touchscreen is also not working - if the touch works but not the display it can be easier).
