I used to have this issue where a few seconds after Skype started, it would just quit. Well, after going through the Skype forums, I found a solution that works for me.
Just add the a new folder called Logs in the .Skype folder in your home folder. It should look like this: \home\[user name]\.Skype\Logs
That is it. Skype may force quit by itself once or twice after that, but eventually it will stay running like it's supposed to. Hope this helps out some out there.