Installing Battlefield 4 on Linux is easy. Right click the game in your library and go to "Properties...".
Then under the compatibility section enable "Force the use of a specific Steam Play compatibility tool". Choose the latest Proton Version.
After closing the properties menu you will be able to download and install Battlefield 4 over the normal "Install" button.

There are 3 free DLCs that contain more maps:

• Legacy Operations
  
• Night Operations
  
• Community Operations

Go to the DLC Page of BF4 and add/download them to play all maps.

We now need to update PunkBuster to make sure we don't get kicked from online games.
Since we're on Linux this can be a bit tricky but following this guide should make it easy.

Protontricks is a utility tool that makes launching external Windows executables inside Steam game environments easy.
Make sure it is installed on your Linux System.
You can find how to install it for your distro on the Protontricks github.

Now we need to download the newest Windows version of PunkBuster. This is important, because Protontricks uses the Windows version of PunkBuster to run Battlefield.
To download PunkBuster go to their website.
Unzip the downloaded file to access pbsetup.exe.

Make sure you have started Battlefield 4 at least once!
To start the installation use the following protontricks command:
protontricks-launch --appid 1238860 ./pbsetup.exe

With this we tell Protontricks to launch pbsetup.exe inside the Battlefield 4 environment. The AppId of 1238860 corresponds to Battlefield 4. The location of pbsetup.exe is the last parameter. Make sure it matches to where you saved the file!

After a moment the PunkBuster Installer Windows should open.
Here you have to accept the License Agreement.
You will see the configuration menu next. Select the "Add Game" Button (marked with red).
Add Battlefied 4 from the dropdown menu. The file location should be found automatically.
Now it will update PunkBuster for your game installation. After this, Battlefield 4 shows up in the menu.
You can now close out of the menu by pressing the "Quit" button.

If you're having trouble with running Protontricks this section might help you by detailing common errors.
When starting Protontricks you might get the Error "SyntaxError: Invalid file magic number".
Make sure Protontricks uses the newest (experimental) version by setting the PROTON_VERSION variable:
export PROTON_VERSION="Proton Experimental"
If Protontricks does not find your Steam directory directly you can set the STEAM_DIR variable:
export STEAM_DIR=/path/to/steamdir

If you have any suggestions or have specific problems (with or without solution) please let me know in the comment.
I will add it to the guide.

The main section of the guide was first written by Yagato on the protondb entry for battlefield 4.

Thank you for reading!