With great difficulty. There are guides out there on XDA and on the LineageOS wiki, but in the end you’ll have to figure a lot of this stuff out the hard way.

You can start with a generic GSI image. It’ll probably be a terrible experience with bad power management, missing functionality, and mediocre cameras, but you should be able to get the bare basics to work. This assumes you have unlocked the bootloader and made a backup of all system images, of course.

I’d you can’t unlock the bootloader, you’re screwed. You could try to find an exploit to bypass the phone’s protections, but unless your phone is vulnerable to as known exploit chain, that’ll take weeks or months or reverse engineering and exploit crafting, an may leave you with a bricked phone.

This assumes there’s an open recovery image for your device. If you can’t find something like TWRP for your device, you’ll need to download the TWRP source code and make a version for your phone.

To go beyond GSI, you’ll need to compile Lineage yourself. For that, you’re going to need a Linux computer (or VM) with at least 200GB of storage, 32GB of RAM, and as many CPU cores as you can find. Having more than 32GB of RAM will greatly speed up this process, as will storing everything on an SSD. It’s probably best to first learn how to compile a known good image (i.e. an emulator image) so you get to know all the steps involved.

If you want an actual install, you’ll need to make a port for your device. Look for a supported device with similar specs, and use that as a base. Get the kernel sources from the manufacturer (which they legally will have to provide, but good luck getting those from certain brands) and configure the kernel build in a way compatible with the Android boot process. Extract all the drivers and such from the backup you’ve made and put them in the right place. The LineageOS wiki has a guide for that. Then build Android on your machine, flash it onto your phone, and see if it boots. Compiling Android can take between half an hour and two hours depending on how fast your computer is. Close down applications if you can’t dedicate enough memory to it, because it will die and bail out if you run out of RAM halfway through. If you’re stuck with a hard drive for storage, expect compilation to take a long time.

You’ll likely need to tweak some SELinux contexts and configure boot scripts to get everything working perfectly. When all is said and done, you’ll have your own build of Android that you can do whatever you want with.

Note that if you publish your final image online, you’ll need to provide kernel sources, as well as other source files you’ve altered, depending on the license they’re under. You’re not allowed to redistribute the binaries you’ve extracted from the phone, though!