I don't actually use Unlocker, I use a similar one called WhoLockMe (I didn't suggest it because it doesn't state that it operates in Windows 7) and it has to be run once to install - after that you right-click on the file in question and WhoLockMe is there in the pop-up menu above "Send To" and some other stuff (every menu will differ some, depending on what other programs you have installed.)
However, on the Unlocker web site, it says "Unlocker Assistant stays in your system tray bar and automatically launches Unlocker if you are trying to delete/rename/move a file that is locked. So for those who use Unlocker Assistant, no need to right click files or folders anymore. This feature can be deactivated during the install." From that, it sounds like, assuming that you have Unlocker Assistant installed, Unlocker should start automatically when you have trouble with a file.
So see if you have Unlocker Assistant in you system tray bar. And, yes, that would mean it kind of runs in the background.