I will keep this guide quite simple.
Step by step how-to Install iOS 5:
Connect the device to the computer and open iTunes.
Click Check for Update
Click Restore button
Click Help → Check for Updates on iTunes menu to get iOS 5 software update.
Click Next, Accept Agreement and iOS 5 installation will be started.
If somehow the iOS 5 software update not showing-up, you can download iOS 5 IPSW directly from Apple’s server and save the IPSW file to your computer. Download iOS 5 IPSW file.
Then, click Shift+Restore button on iTunes and select the IPSW file which you just saved on the computer. The iOS 5 installation will be started then.