Step 1: First off, you will need to root your Galaxy S III, and then flash ClockworkMod Recovery to it.
Step 2: Download the latest nightly of CM10.1 from the CM Downloads page,
as well as GApps for CM10.1, which you can grab by doing a simple
search on your favorite search engine. Once downloaded, drop them into
the both into the /sdcard directory of your handset.
Step 3: Next, you will need to shut down your Galaxy
S III before booting into ClockworkMod Recovery. If you’re unsure of
how to do this, you just need to remember to hold Volume Up + Home + Power simultaneously.
Step 4: Take a Nandroid backup.
Step 5: Next, you’ll want to navigate to install zip from sdcard > choose zip from sdcard and then select your CM101.zip file. It’s best to rename the CM 10.1 zip you downloaded in Step 2 to CM101.zip so that you can locate it easily.
Step 6: Apply the same process as in step 5, but for GApps.
Step 7: From within the ClockworkMod main menu, select wipe data / factory reset, and once the process is complete, opt to reboot system now from the main menu.