Since this is not an official ROM, therefore you will need a ClockWorkMod installed on your device, this custom recovery is needed in order to install the CM10.1 and the Google apps.
Interested? Good. But please read the following first.
1. Backup your Android data first.
2. Have a rooted Galaxy S3 SCH-I535.
3. Enable USB debugging mode of your Android's OS based on your existing Android version, if it's necessarry.
Remember, you are invited to flash this ROM but you are not forced to do so. So since the decision is yours to take, so its the responsible of any results installing is also yours. Thus blame none but yourself, if things go wrong!
How To Install CM10.1 ROM (Jelly Bean 4.2.2) on Verizon Galaxy S3 SCH-I535
Step 1 -Download CM10.1 Android 4.2.2 ROM and download Google apps to your computer and store it in the place you can easily find like Desktop for example.
Step 2 -Now plug your smartphone to the PC of yours using a USB cable.
Step 3 -Then copy this Android 4.2.2 ROM zip and paste it to your SD card. Place the files in the SD card root.
Step 4 -Now shut down your phone, then unplug your device from USB.
Step 5 -It is time to enter the Recovery mode. Simply turn your device ON at the same time, press and hold these buttons simultaneously: Volume Up + Home + Power.
Step 6-And within the ClockworkMod recovery, do a full data wipe first. Then find this “install zip from SD card” then click and hold the Power button to select it.
Step 7 -Again press Power then choose “choose zip from sdcard”.
Step 8 -Use the volume keys to go to the Android 4.2.2 zip you pasted to SD card, then choose it using the Power button. Then on the next screen confirm the installation for it to begin. (Do from step 1 to step 8 again to install Google Apps).
Step 9 -Once the installation is completed, head to “+++++Go Back”. Then reboot your device, by choosing the “reboot system now” on the recovery. Your device will take reboot, this process may take few minutes, therefore just wait.
If you successfully followed the above steps, that means your Verizon Galaxy S3 is now running on CM10.1 ROM, to make sure then head to Settings -> About to check what OS is running there.