Black screen

I can't tell you what you would need to do because neither you nor I know the precise problem. If it's the display, then you need to find another one. Parts will not be easy to find, especially the display. I have no idea where all those parts would be. There are many sources out there including eBay. You would need to do some searching. The display will cost around $600, if you can find one. And, it's a devil to replace. The GPU is socketed, so if you can find the chip, then replacing it isn't too hard once you take the machine apart. In all, the cost of parts plus your time and labor are worth far more than the machine is worth.

You could try hooking up an external monitor. If it works, then you can offload your files to an external drive. You could disassemble the computer by removing the display, then remove the HDD. Put it in an external USB enclosure to connect to your new computer. If you can find a working TB3 to TB2 adapter, then you could hook up the two via TB cable, boot the iMac into Target Disk Mode and access the drive from the new computer. If you can get the iMac working with an external monitor, then you can also transfer data via LAN.

You might want to try this:

https://support.apple.com/en-us/HT210905

I don't think you will find much help. You probably lost the display and/or the GPU. If that's the case, then you are on your own for repair. Apple no longer makes parts, and Apple Stores will not service obsolete hardware.