I would like to root my Verizon Samsung Galaxy S4 SCH-I545.
It is currently running Android 4.2.2 and has Build number ending in VRUAME7.
How do I root my Galaxy S4?
If you are lucky enough to have not updated and are still on the VRUAMDK Build number then here is the root method (source & credit given to: http://forum.xda-developers.com/showthread.php?t=2290798):
STEP 1: Go to the source link above and download the five items in the first step, Samsung Official USB Drivers, ODIN v3.07, VRUAMD2 Prerelease Kernel, Stock VRUAMDK Kernel for your phone, and Motochopper Root Exploit Tool.
STEP 2: Boot your phone into ODIN mode.
STEP 3: Flash the VRUAMD2 Prerelease Kernel using ODIN v3.07 in PDA section only. Boot back into Android and verify that the flash was successful.
STEP 4: Follow Settings -> More -> About phone -> tap "Build number" 7 to unlock Developer options and verify in there that you have "USB debugging" checked and "Verify apps via USB" unchecked. Verify in Security setting on your device that you have "Unknown sources" checked and "Verify apps" unchecked
STEP 5: Connect your phone to your computer and open "Run.bat" file from the location you have extracted motochopper to. You will see a message on the phone asking you to accept the new computer connection (select to always allow and select OK).
STEP 6: After motochopper runs and your phone reboots on its own, check if SuperSU has any update in the Play Store. Open SuperSU app and let the binary update itself to the latest VERY IMPORTANT OR YOU WILL LOSE ROOT ON STEP 4. Now you can open a root application and verify that is able to access root properly.
STEP 7: Now to restore the stock kernel you need to boot your phone into ODIN mode (how-to link above).
STEP 8: Flash the Stock VRUAMDK Kernel using ODIN v3.07 in PDA section only (follow same link in step 3 for a how-to).
STEP 9: Boot back to Android and verifying everything is still working and kernel has flashed successfully. You should now be fully rooted and on the fully stock original firmware that came on your device.
To restore a custom recovery (optional), follow Step 5 on the source link.
There is a root method available for the VRUEMA7 Build that does not require you to wipe your device or flash ROMs, but it does not have an unlocked boot loader (source & credit to: http://forum.xda-developers.com/showthread.php?t=2380325)
STEP 1: Download all three files from the first step from the source above and install/extract as necessary.
STEP 2: Insert a microSD card (exFAT format REQUIRED) into your device and connect it to your computer as a media device.
STEP 3: Move the extracted files and folder from the Root Exploit package onto the root of your microSD card. Do not put them in ANY folder.
STEP 4: Open the Android Terminal Emulator App on your device and run the following commands hitting return after each line:
cd /mnt/extSdCard ./pwn ./script.sh
After the second command you should see the '$' change to a '#' indicating root access. The third command will make root access permanent.
STEP 5: After the above commands have run, you can download and open SuperSU and check for binary updates. At this point you should be done and you can open an app that will request root access to test this.
Here you will find a guide for the stock ROM version of your Galaxy S4
However, if your phone was updated to the VRUAMDK, you should use the already posted tutorial. Good luck!