The procedure is relatively simple but pay attention to version numbers for the various software involved. Steps would be:
- Download OmniROM and GApps packages (.zip) to the phone.
- If not rooted, root the phone and flash TWRP recovery.
- Flash OmniROM and GApps packages, wipe data(!), reboot
After upgrade, I had all the apps and their data remain from the 4.3 version.
- Omnirom download page
- Changelog for the next nightly
- GApps page at XDA
- TWRP Recovery for S3. Download from here (link). The version of TWRP that works with Android 4.4 KitKat is 126.96.36.199! Pay attention, as older versions will result in metadata error, leaving your phone with wiped out /system partition, meaning no OS to boot. Download the .tar file and flash with Odin.
Also, some first-aid type of texts:
- Odin versions (link)