Although Alexa is a flawed metric (based only on visitors who have their toolbar/widgets installed), it may still be a web metric that indeed does affect website rankings and PR. This is because Alexa has someone come to be known as an authority site, especially in the Internet marketing industry. Again, it's is a flawed biased metric no dount, but if you are dead set on improving your Alexa rank then you might try doing some of the following:
install Alexa widgets on your site (even the review widget will be fine)
install Alexa toolbar (and for best results visit your site with different IPs)
Join networks where people have Alexa toolbars installed (yes they do exist)
Pay for Alexa traffic (I wouldn't bother but you can if you want)
Keep in mind that no matter what you do to improve your Alexa ranking, it will only be based on stats over a 3-month period of time. Therefore, you will have to keep repeating Alexa boosting methods every few months if you want to keep your Alexa ranking on top.