1. Download

Download the client.

2. Extract

Once downloaded, extract everything to your "steamapps/sourcemods" directory. You should now have "steamapps/sourcemods/tf2classic".

3. Install Source SDK Base

Go to the 'Tools' section in your Steam client and Install Source SDK 2013 Base Multiplayer.

4. Restart Steam

Once it has finished updating, restart Steam

Check your games. You should see "Team Fortress 2 Classic" has appeared on your games library.