Time Machine backs up new files every 60 minutes, so would very often show "In Progress"
If the backup complete message comes up, you can then erase your active hard drive, with the possibility that you may lose some document that you have modified within the last hour. It's very likely that won't affect you in any significant way.
So, now that you have your backup, you ask "what needs to be done next"?
next, is boot to your recovery system. (Restart, holding Command-R)
Run Disk Utility in the recovery system. Select your boot partition (it will be named, such as Macintosh HD, or something else if you, or some previous owner, changed the name. Click the Erase button. You can name your erased partition something else - otherwise it will be named the default name "untitled", I think. You can make the name anything you like.
Click Erase to continue. It only takes a few seconds. When complete, you can Quit Disk Utility. That will return you to the main menu screen. Choose Reinstall OS X.
You have to choose the destination for your OS X install, which will be the partition that you just erased.
Continue with that. The system files will be downloaded over your internet connection, then continue with the OS X install. May take from 30 minutes to a couple of hours. It restarts automatically to complete the install.
Wait some more - there will be a progress bar of some kind. Don't be concerned if nothing seems to happen for some time. You may notice delays of 10 to 30 minutes (or not), so just let it go, do something else for a while.
And, Finally, you will see a screen asking if you want to migrate files and settings from somewhere else. One of those choices will be "Time Machine Backup" (or something similar to that), and that's when you want to make sure that your external drive is connected. The system will then look for that backup, and you just continue on, the migration tool will get all the files and settings that is needed to get the user back into place, and all your files and apps, too.
Easy - there's just lots of waiting.