What version of avast are you using Home or pro ?
Home I don't believe should be a problem, you need to have kept the registration key, also if it complies for Home, non-commercial use then you can have multiple Home versions in the same house using the same key.
Pro, I'm not sure about (I don't work for Alwil, so can't speak for them) but I thought it was a single user license not restricted to a single PC providing you uninstall completely from the existing system and install clean on the new one using the existing registration key.
Hopefully one of the Alwil team will be able to confirm this if you are a Pro user.
