Well first you want git, to get it you just open up a console and do:
sudo apt-get install git
This is also how basically everything on Linux is installed including the libraries you'll need.
Then you want to clone from the git onto your computer.
To do this you'll want to first use the command line to navigate to where you want the working project to be, then do:
git clone https://github.com/CleverRaven/Cataclysm-DDA.git
Next you'll want to install all of the libraries, I suggest you look at this:https://github.com/CleverRaven/Cataclysm-DDA/blob/master/COMPILING.md
Keep in mind you'll also want the dev versions of these libraries, for example instead of just sdl2_mixer or libsdl2, you might also need sdl2_mixer-dev or libsdl2-dev.
I don't remember exactly which ones you need, but they should be listed on the github.
Once you've done that, you'll want to try to compile it, so navigate to the Cataclysm-DDA folder with the Makefile in it and try something like:
MAKE TILES=1 RELEASE=1
At this point you'll probably get an error about some library you missed, so just run an apt-get at whatever the error was about.
Remember to try appending "-dev" on the end of it if it doesn't fix itself with a simple attempt.
You can compile with a bunch of different options, for reference I think my normal compile command is something like:
MAKE TILES=1 SOUND=1 LUA=1 RELEASE=1