I had the chance to peep into how an Ubuntu package is cooked and submitted: the procedure is clearly described in clear, detailed steps on the Ubuntu wiki, so this is just a summary for my own reference: I'll give a lot of details for granted.

First of all you need a Launchpad account, with your SSH public and GPG key:

http://packaging.ubuntu.com/html/getting-set-up.html

Then you need to setup your workstation for building packages and various tooling for patching package sources:

http://packaging.ubuntu.com/html/fixing-a-bug.html

I don't like adding a lot of random packages to my workstation, so I wrote down all the required packages so I can replicate the environment in a virtual machine or (as Ubuntu suggests) an LXD container - although I'll investigate if it can run GUI applications.

Now, I discovered there are two ways to patch the sources of a package. One involves using edit-patch (1), as described in the previous link, the other revolves around quilt (1), a nice tool that Andrew Morton wrote for his own purpose of maintaining patches to the Linux kernel (the -mm branch). Usage is detailed here:

https://raphaelhertzog.com/2012/08/08/how-to-use-quilt-to-manage-patches-in-debian-packages

and

https://wiki.debian.org/UsingQuilt

edit-patch is easier to use, but if you feel like playing with quilt it'll only take 5 minutes to get acquainted. So, basically the procedure consists in:

1. ensure to have deb-src packages activated in your sources.list, then run apt update
2. create a patch containing all the changes that you want applied to the package (e.g. git diff > patch.diff)
3. create a work directory somewhere and cd into it
4. get the sources for the package you want to update: pull-lp-sources <ubuntu_release> <package_name>
5. descent into the source directory and either use edit-patch or quilt to create and apply the patch you create before
6. sign-off the patch and fill up some paperwork (changelog and other stuff)
7. locally build the package, install it and test if it works
8. submit your patch (a .debdiff file) on Launchpad, attaching it to a bug. Await comments from the maintainers.

For any question or doubt, support is available on irc.freenode.net on #ubuntu-motu or ubuntu-motu@lists.ubuntu.com.