I have had some problem with Avast. When all protection is enabled, Skype automatic shutdown. I can’t connect to Skype.
I can connect to Skypes site, but not download or logging in to Skype.
I thought it was my Internet connection. So I disabled the VPN. It still had issues, so I disabled Avast. Now did everything work again.
I use Windows 7 64bit and Avast Premier. I really want to use Avast, but I can’t use it when I use Skype.
All you have to do is setting up the firewall correctly.
You can verify that it’s the Avast firewall that’s the problem by disabling it.
If that fixes your problem, follow Eddy’s advice. 
good advice from bob. simply disable the avast firewall. if you then can connect you need to add skype to the firewall as eddy said. if you do not know how to do that please ask and we can walk you through it. if you still can not connect after disabling the firewall then we can look a bit further to see why.