It is very likely your camera actually works out of the box. Install a program that uses it such as skype
or cheese
.
See if the camera works
sudo apt-get install cheese cheese
If everything is OK, that should show you a live feed from the camera.
Install video4linux control panel to be able to tweak the settings
sudo apt-get install v4l2ucp v4l2ucp
Seriously though, you could have found this by clicking on the first google search result for "linux mint logitech cam". In the future, please try looking around by yourself a bit before posting here.