I think you want to DECREASE Alexa rank, not increase it
(the lower number the better)
Aside from other techniques, you may use your own visits to your site (need to have Alexa toolbar in Firefox activated) - just while you working on your site - adding blogposts, reading comments, editing etc. - everytime you visit your own page, you decrease Alexa rank a bit. Note that results can be observed after a day or sometimes some days later.
Please note that visiting your competitors sites with Alexa toolbar enabled, you improve their rank too.