Here's how to root Verizon's Samsung Galaxy S3 that already receives the official Android 4.1.1 Jelly Bean OTA update build I535VRBLK3. Rooting the Verizon Galaxy S3 has never been too easy as it may seems like other U.S Galaxy S3 variants - AT&T SGH-I747, T-Mobile SGH-T999, and Sprint SPH L710. The Verizon Galaxy S3 involves bootloader locking and unlocking requirements, rooting your device with other means will surely ended up into bricks that is hardly enough to recover. So to root it, you need a delicate but yet simple procedure before you can actually install the SuperSU apps that fully access root onto the device.
Now head over to XDA-Developers to get the needed files to root your Verizon's Samsung Galaxy S3 that runs on Jelly Bean. Just give credits to open1your1eyes0 and the rest of the gang involved in making this possible rooting technique - http://forum.xda-developers.com/showthread.php?t=2046439
Here's a visual aid for beginners that might confused from the advance users guide.
Flashing the Bootchain with OdinThe first bootchain is VRALEC, you need this so that you can install the Custom recovery of your choice like CWM (ClockworkMod Touch Recovery 18.104.22.168) or TWRP (TWRP Recovery 22.214.171.124).
Assuming the you already installed the USB driver. Now run Odin then click on PDA button then browse to the downloaded VRALEC.bootchain.tar.
Boot the Galaxy S3 into Odin Download Mode
Hit Odin's "START" button to flash the VRALEC bootchain. Just wait until it finishes.
Flashing the Custom Recoveries CWM or TWRP with OdinYou need to choose the custom recovery of your choice, either CWM or TWRP. You will need this so that you can the install the SuperSU apps via SD card update zip option.
Download the TWRP or CWM onto your computer.. Now load the CWM or TWRP tar files onto Odin's PDA button. This time Uncheck the "Auto Reboot" checkbox option.
Flashing with CWM - load the CWM6012TouchVZW.tar.md5
Flashing with TWRP - load the TWRP2310VZW.tar.md5
Now boot the phone into download mode again, then hit "Start" to flash the custom recovery. Remove the phone from PC once finishes.
As soon as flashing recovery is done in ODIN and you see the screen go black, immediately press and hold Home+PowerUp before the device powers on again (have to do it quickly) and hold it until you see it boot into the CWM/TWRP screen.
Install ROOT or Unlock Bootloader
Download one of the below
For Root Only - Download SuperSU to your SD Card - Link
For Unlocked Bootloader Only - Download Verizon Galaxy S III Unlocked Bootloader to your SD Card - Link
For Both Root and Unlocked Bootloader ***RECOMMENDED*** - Download SuperSU + Unlocked Bootloader to your SD Card - Link
Restore the Original Jelly bean VRBLK3 Bootchain
This step is optional but recommended to return your original partitions after you see everything is working properly
Boot phone into ODIN mode
Flash the VRBLK3 Bootchain using ODIN v3.07
After a final reboot, you are all set to use your device as needed