There are guides on the Khadas docs page for Linux (the ones that say “via CLI” or “via Command Line”). Nothing requires Windows except for the Amlogic USB Burning tool, which is only really useful for the Android ROMs (won’t even work for most of the other ROMs).
The easiest way is using bootable SD cards. The guides tell you to use dd(1) command but I recommend Etcher for writing images to SD card (Linux version available, very easy to use, very reliable and good quality results).
You might need to write Uboot to the SD card before using Etcher (you can find the uboot files on the Firmware page):
sudo dd if=u-boot.bin.sd.bin of=/dev/sdb conv=fsync,notrunc bs=1 count=444
sudo dd if=u-boot.bin.sd.bin of=/dev/sdb conv=fsync,notrunc bs=512 skip=1 seek=1
I’m not sure which images this is necessary for, though.
Booting from USB drive apparently also works but I haven’t tried that yet.
Most of the imgs I’ve tried from these forums work like Linux desktop Live CDs: you boot them, try them out and then install to internal memory from within the booted operating system. E.g. on the Ubuntu ones you run ./install.sh and on the Libreelec ones you choose “install to eMMC” from the shutdown menu. You will need to follow exact instructions from the relevant threads.
Booting from an SD card can be a bit tricky but I’ve used the MRegister method successfully when nothing else works.
Another method mentioned in these forums is using the Android 6.0 ROM, going into the Update&Backup app (Settings->Update or something like that). Then you choose aml_autoscript.zip from an img burnt to an SD card and apparently it can use the Android OTA update system to apply an update. This hasn’t worked for me.
Finally, advanced users can directly interract with Uboot using UART or TFTP. You can do lots of things with this, I guess, including flashing images, booting in different ways, etc. There are guides for that on the Khadas docs page, but you will probably need to use external resources to learn about Uboot.
A word of warning about Ubuntu on the Vim: graphics support is not very good but hopefully getting better.