Popular blogger Maki at doshdosh.com has come out with 20 different ways to improve one's Alexa rank.
Some of them are quite obvious and straight forward like ...
1. Install the Alexa toolbar
2. Put up an Alexa rank widget
6. Write or Blog about Alexa
While there are few, well thought innovative ones ...
9. Use Alexa redirects
10. Post in Asian social networking
18. Pay Cybercafe owners
Maki's post contains a very good introduction about Alexa Rank and why would someone be interested in it? The blogger has also explained it simple terms on how to get started with Alexa. There is also section which talks about the problems and credibility of the ranking system.
Check it out for more details