I should first point out that there are other options to unlocking yourself, since this option is a bit more involved and can seriously mess up your phone:
1) You can get the unlock code from your provider for free if they are prepared to give it to you. If they are then that is the safest and best option.
2) For $25, you can use TheMotoGuide's unlocking solution
which is instant and guaranteed.That being said, here's directions on how to unlock the phone yourself:
*You need to downgrade your boot loader to 7.D0, be aware that this can be risky.
* You may find afterwards that when starting in bootloader you have only a blackscreen, so long as the phone works ok and you can connect in bootloader that is nothing to worry about
* Ensure your phone is fully charged before starting. I recommend before beginning you first make a backup of your phone as described in the back it up
section. Especially the PDS back up. This assumes all necessary drivers are already installed.
1) Start in bootloader mode (hold # and * while powering on)
2) Identify your current bootloader (e.g. 8.26 or 7.E0)
3) If you already have bootloader 7.D0 skip to step 6
4) Download bl_826-828_to_07d0_for_V3_by_Archy.V2
5) Flash with this file using RSD Lite (described at the Flashing and Flexing section)
6) Download and extract the MOTO UNLOCKER SOFTWARE
7) Start in bootloader mode (hold # and * while powering on). In bootloader mode after the downgrade it is not unusual for the phone to show a blank black screen with no info, don't worry, though, it is still in bootloader mode!
8) Open up Moto Unlocker and connect you phone
9) Run Moto Unlocker click connect and yes! Once finished it will look like this:
Congratulations, your Motorola Razr V3 is now unlocked