If my old Mac computer screen doesn't work what can I do to recover my files?

You can connect an external display, although it can sometimes be hard to get the external display activated. You may need to log into macOS blindly before you can use the external display. To log in blindly wait until the screen should show the login screen. Then type the first character of one of the user accounts and press the "Return" key. Now type the password and press the "Return" key to submit the password. If successful, you should see something on the external display. Try clicking on the menu bar on the external display. I find it useful to then click the Apple menu and select "System Preferences". Go into the "Display" pane and select the "Arrangement" tab (or you may first need to click the "Gather Windows" button). Within the "Arrangement" tab, select the "Mirror Displays" check box.

If this is a laptop you can also try putting the laptop into Clamshell mode. You will need to have the charger connected, plus you need a wired USB keyboard and mouse connected. Immediately after powering on the laptop, close the lid and hopefully you will see macOS booting using the external display. I find triggering Clamshell mode can be quite difficult sometimes.

Another option is to put the laptop into Target Disk Mode which will make the laptop act like an external drive. Use another Mac to connect to the laptop which is Target Disk Mode. It doesn't matter if you cannot read the screen on the broken Mac. In theory the other Mac should automatically mount the drive of the broken laptop or it should prompt you to mount it. However, if macOS does not mount or prompt you to mount the drive, then launch Disk Utility and manually mount the drive of the broken laptop. If the broken laptop was running macOS 10.13+, then your other host Mac will need to be running macOS 10.12+ in order to see the APFS volume.

If the laptop has a removable SSD and it is an Apple original SSD from a 2015 or earlier Mac (or a MBAir 2017), then you could remove the SSD and install it into an OWC Envoy Pro Enclosure so you can use another Mac to access the data. This enclosure is expensive. If you have an Apple USB-C Mac, then you cannot remove the drive, so this is not an option.

FYI, it always helps to provide the exact model of your Mac as well as the current version of macOS which is installed so we can cater our answer directly to you and save us some time making a generalized response.