You can write to Indiana State Police and inform them of your new address (return receipt requested, of course), or I have heard that you can just call the licensing division at the number on the website (if it is still listed there).
You don't need to request an updated copy of your license to be sent to you ($20.00 for a new copy), but you can if you wish. And if you go out of state with your firearm, someone mentioned a while back that that their LTCH and Drivers License addresses needed to match.
If you're talking about updating your address with INGO, there is no such animal to update.
i just did this a couple days ago if you're talking about your LTCH. there is a form online at in.gov that you fill out and print off. I just faxed mine in and was good to go.