CUSTOM RECOVERY:

So most of you guys who've seen the previous post on "How to root the international GS3" need to have a custom recovery as it offers more options than the stock recovery and you can use it to flash software to your device instead of using ODIN downloader, so basically installing custom recovery is the same as rooting your phone [Check previous post] the only difference is the file you're going to flash isn't the CF-Auto Root but it's going to be the custom recovery itself.
There are lots of custom recoveries out there, but this is a very good one called Clock work Mod (CWM) which has a pretty good list within the interface, the recommended one is 6.0.1.2 version as most ROMs are compatible with it. DOWNLOAD CWM 6.0.1.2 BY CLICKING HERE.

What CWM offers:
1. install zip from SD-card
   a. choose zip from sd card
   b. apply /sdcard/ update
   c. toggle signature verfication
   d. choose zip from external sd-card

2. wipe data/factory reset
3.wipe cache partition
4.backup and restore
    a.backup
    b. restore
    c. delete
    d. free unused backup data
    e. choose backup format
    f. backup to external sd card
    g. restore from external sdcard
    h. advanced restore from external sd card
    i. delete from external sd card
    j. create image.zip

5.mounts and storage:  mounts and formats: cache/system/data/preload/sdcard/externa_sdcard you choose everything seperaltley.

6. advanced menu
  a. reboot recovery
  b. wipe dalvik cache
  c. wipe battery stats
  d. report error
  e. key test
  f. show log
  g. fix permissions


so almost everything is included, and the 6.0.1.2 is also a touch screen support: you don't need to press the vol up/down home/power to navigate through..
the download link is here.
The download process as I mentioned earlier is pretty much the same as rooting your phone-check previous post- but you would rather flash the file I've linked in here.

How to root the International Samsung Galaxy SIII?

  Here you'll find the answer dude, so whether you are sick of the TouchWiz stock look and want to get a new software raockin' your phone or you just want to download some Apps that are not available in your country  or just want to test some stuff you need to root your phone to be able to do such things.
Firstly by rooting your phone you are voiding your warranty, and you are increasing your flash counter.
Q: What's a flash counter?
A: when you download unofficial firmwares to your device you are increasing that counter, which is made by Samsung to be able to know how many times you've flashed unofficial firmwares to your device or to know if the current firmware is Samsung official or not, but thanks to XDA's elite recognized developer Chainfire who developed an app called triangle away that resets that counter, more info here.
                                                                                         
                                                                                                   Flash Counter in download mode:

 Further on, there are many ways for rooting your device the first is a very easy-way rooting via SAMSUNG GALAXY S3 TOOLKIT and the other common way is by CF-auto-root by Chainfire, thread here.

We will include Chainfire's method, this also works not only for the international version but other versions, we will only include the international SGSIII, if your using another device return back to the original thread.

1. Download the root for i9300 from here.

2. Download ODIN from here.

3. After downloading both files, extract odin and launch it.

4. The screen should look like the image and make sure that nothing is checked but "F. Reset time" and "Auto Reboot" just like the image below

5. Now check the PDA icon and choose the file you've downloaded from Chainfire's thread which has an extension of .tar.md5

6. Now, just put your phone into download mode by shutting it down and opening it again however while opening again press power + volume down + home key, make sure you have the latest Samsung USB drivers.

7. After you've connected the phone when it is in download mode you should have a screen as follows:

8. Now press start sit back relax and it will take like 10-20 seconds and you're phone will reboot after that remove the usb and here you go you've a rooted SGSIII!