If you don't already have Steam installed, download it here. If you don't have an account, signing up is easy, and you don't have to make any purchases to play GoldenEye: Source.
Also necessary is Source SDK Base 2007. Steam will install this for you if you don't have it, but you can get a head start by clicking here. You can also find it in the Tools tab of the Steam library.
Download links for GoldenEye: Source can be found on the Download page.
If you downloaded the installer: After the download completes, simply run the installer and follow the instructions.
If you downloaded the zip file: place the gesource folder inside
Steam/steamapps/sourcemods. Note that you can only install Source mods to your main Steam library (under the location which Steam itself is installed). Make sure you install the Visual C++ 2012 Redistributable. Get the vcredist_x86.exe file, even if you are using a 64-bit operating system.
For GoldenEye: Source to appear in your games library, Steam must be restarted after the installation is completed.
Now that you've made it this far, you are ready to play GoldenEye: Source!
If you encounter any issues, check Client Troubleshooting.