[General Questions and Answers] jamesclassy:
Hi this is how to root your s4.

Pre-requisites

1) Install USB Driver for Samsung Galaxy S4 to transfer files between the Android device and the computer.

2) Enable USB Debugging Mode on phone to connect it with computer and use Android SDK with it. [Press Menu>> Settings>> Applications. From there navigate and click on Development option and check USB Debugging Mode.]

3) As the tutorial might wipe out all data of the phone, create a backup. If the device is having any custom recoveries such as ClockworkMod/TWRP already installed, backup data using the recovery.

4) The Galaxy S4 should have above 80% battery charge.

5) The phone should be factory unlocked and not locked to any particular carrier.

6) The device will lose manufacturer's warranty once it is rooted. However, users can reclaim the warranty back by unrooting the handset later.

7) The CF-Auto-Root files work only on Samsung Galaxy S4 GT-I9500 and GT-I9505. As applying these to any incorrect variant might brick the device, verify the model number of Galaxy S4 by navigating to Settings>> About phone>> Model number and install the root package accordingly.

Steps to root Samsung Galaxy S4 I9500/I9505 on Android 4.3 Jelly Bean

Step-1: Download CF-Auto-Root file for Galaxy S4 to computer (Download link download.chainfire.eu/316/CF-Root/CF-Auto-Root/CF-Auto-Root-jflte-jfltexx-gti9505.zip

Step-2: Extract the downloaded root zip using Winzip, 7zip, winrar or any other extracting tool to get a file with .tar.md5 extension

CF-Auto-Root-jflte-jfltexx-gti9505.tar.md5

Step-3: Download Odin3_v3.07. Extract the Odin zip to get Odin3 v3.07.exe file

Step-4: Switch off Galaxy S4 and wait for five to six seconds for vibration to confirm complete power off

Step-5: Place the phone in Download Mode by pressing and holding Volume Down, Home and Power buttons together until the construction Android robot icon with a warning triangle appears on the phone's screen. Then press the Volume Up button to enter Download Mode

Step-6: Run Odin3 v3.07.exe as Administrator

Step-7: Connect the phone to the computer using USB cable while it is in Download Mode. Wait for some time until Odin detects the device. When the phone is connected successfully, an Added message will appear under Odin message box. Additionally, the ID: COM box in Odin will turn blue with the COM port number indicating successful connection

Note: If the Added message does not appear, then try another USB port, try the one located at the back side of the computer. In case Odin still fails to recognise the device then reinstall USB driver.

Step-8: In Odin, click the PDA button and select CF-Auto-Root-ja3g-ja3gxx-gti9500.tar.md5 or CF-Auto-Root-jflte-jfltexx-gti9505.tar.md5 file according to the variants

Step-9: Ensure Auto-Reboot and F.Reset Time checkboxes are selected in Odin

Step-10: Click the Start button in Odin. The installation process should begin now and would take few minutes to complete

Step-11: When the installation process is completed, the phone will restart and soon a PASS message with a green background in the extreme left box at the very top of Odin. Once the home screen appears, unplug the USB cable from the phone to disconnect it from computer

Samsung Galaxy S4 I9500/I9505 running Android 4.3 should now be rooted successfully. You can now install any app that requires root permission. To verify the root method is installed and working properly, install Root Checker app from Google Play Sore.

Press "Thanks" if it helps :)