Ubuntu recently added the ability to customize this without relying on any third-party scripts (this is not yet possible if using GDM with a distro other than Ubuntu, see here for progress on that).

First install systemd-container:

sudo apt install systemd-container

Now we need to login as the gdm user before we run any gsettings commands:

sudo machinectl shell gdm@ /bin/bash

Now we can set the background to any image:

gsettings set com.ubuntu.login-screen background-picture-uri 'file:///usr/share/backgrounds/warty-final-ubuntu.png'

Other than background-picture-uri, you can also set background-color to just change the color without setting an image. If you set an image, you can also set the background-repeat and background-size settings.

(Note that you'll only see this image/color on the login screen, not the lock screen, which will instead be a blurred version of your desktop wallpaper)