What AV did you have installed before avast and how did you remove ? also make sure to uninstall any third party firewalls, you can find removal tools here for previous AV http://singularlabs.com/uninstallers/security-software/
please read the info supplied before running removal tools and reboot after running.
Download the latest version of avast 6.0.1367 and save http://files.avast.com/iavs5x/setup_ais.exe
then uninstall current avast version reboot and install new - add the license key you recieved via email and reboot again, should be working now ?
If previous AV's have been properly removed you can try a repair of avast via remove programs - double click avast and scroll down to repair and accept, reboot when finished, if repairing doesn't work then revert to the instructions above.