In order to completely remove the software from your computer, press Alt + Ctrl + Delete key combination, open Task Manager and close all instances of Avast! Internet Security. After that, you can safely remove it through Add or Remove Programs.
Alternatively, you can uninstall the antivirus using Avast Uninstall Utility. Just download the program on your computer from the official web page, reboot the computer, press the F8 key to enter in Safe Mode, launch the uninstaller and press Remove.