You can buy cards in the beta, though you won't be able to keep them when the game goes live. However, you will be given a credit equal to the amount you paid for the cards.
Read more here:
http://us.battle.net/hearthstone/en/faq/
You can sign up for the beta by checking the box next to Hearthstone here: https://us.battle.net/account/management/beta-profile.html

Signed up for the beta a while ago so I hope I get a key. Interesting how they handle the beta wipe though. I imagine that may anger some if they obtained some super rad cards.
I didn't realize you could sign up for the beta, or if I did I had forgotten. I just signed up earlier today, so if there's any chronology to it, I certainly won't get picked.
Yeah, I have no idea how Blizz picks. One time I got beta on a WoW expansion and didn't play it. Haven't been selected since.