I was going to write a large post about how to increase your Alexa ranking but here might be a few things:
Ask friends to install the Alexa toolbar and ask them to browse your website daily.
Download the toolbar on public computers (school, workplace -if possible-, etc. be creative) and make your website as homepage.
Get many webmasters to your page, most of the webmasters have the Alexa toolbar or a plugin that has the Alexa toolbar integrated.
Place an advertisement on your website or make a post on your website about how Alexa can change your visitors' browsing experience and they might give it a try.
Hopefully it helps you a little bit