Required (Windows):

1. Download and install Universal ADB Drivers: https://github.com/koush/UniversalAdbDriver or http://adbdriver.com/downloads/
2. Download and install minimal ADB + Fastboot installation: http://forum.xda-developers.com/showthread.php?t=2317790
3. Open windows command prompt and go to folder when adb is installed - if in doubt SHIFT + right click on a folder containing newly installed adb.exe and select "Open command window here".

Now there are easy way and hard way. Easy first:

4. Reboot the phone into recovery. For Galaxy S3 it's done using Volume Up + Power + Menu combination.
5. Select "install zip from sideload"
6. In adb shell type:

adb sideload C:\YourDownloads\cmXXXXXXXXXXX.zip

7. After installation is successful wipe cache partitions to prevent binary incompatibilities between Android releases:
- wipe cache partition
- advanced
- wipe dalvik cache
8. Optional - you may want to install appropriate gapps package to enjoy google experiece on your phone/tablet: http://wiki.cyanogenmod.org/w/Google_Apps

And a hard way:

4. Open adb shell

adb shell

6. Mount data partition:

mount -t tmpfs none /data

mkdir /data/media
cd /data/media/

7. Open another adb terminal and run:

adb push UPDATE-FILE-YOU-DOWNLOADED /data/media 

8. In recovery select install zip from sdcard -> select the name of the file you pushed to /data/media