Lately, I am getting this Avast pop-up as soon as I start my machine:
I updated the software and restarted, and got the pop-up again. Is it authentic?
When something pops up on my screen that seems doubtful, I sometimes use Process Explorer to figure out which process is generating it. Process Explorer is one of the better freeware task managers out there; you can download it at http://technet.microsoft.com/en-us/sysinternals/bb896653.aspx. There's a button on its toolbar that looks like a target; drag the toolbar button over any window and Process Explorer will highlight the process that created it. If the process that created the questionable window is clearly one of the Avast processes, it's legitimate.
One more thing: the dialog you're seeing isn't telling you to run updates on Avast. It's actually telling you that your subscription to the free version of Avast is expiring, and that to continue using the free version you have to re-register. That explains why the message didn't go away when you applied updates.
Avast has been pushing their priced versions more lately, I'm frankly disappointed in them for doing so in such a spammy way. While I can't guarantee that the screen you're seeing is legitimate, it would appear to be at least similar to legit screens I've seen from avast.
If it keeps nagging you and you're looking for a good free anti-virus product, I've come to rather like Microsoft Security Essentials. It's lightweight and seems fairly effective. And doesn't nag you.
You could always run an anti-malware/spyware scan with something like malware-bytes to make sure it is simply a spammy-authentic pop-up screen.
Also, what you could also try is, click the upgrade link in the box and see where it takes you. If it's authentic, you'll reach avast. If it's fake, you'll be directed to some dodgy URL.
Like nhinkle stated; if it annoys you, go for MSE. It's a very good alternative to any free anti-virus.