Start increasing the backlinks for the website and improve the keyword position for the website which automatically improves the ranking of the website.
First and foremost thing to increase Alexa rank is to write quality content. It will help you to improve website traffic and another thing is Google loves quality Content. Good content always strikes high-ranking in SERP's and will help you to increase the amount of traffic on your site.
Start improving the back-links for the web page and improve the keyword and key phrase position for the web page which instantly increases the position of the web page.
Improve your website, customer oriented things and alexa rank will improve automatically. Do not work for alexa rank, work for your website improvement.
Alexa measures the traffic on your website and provide a numerical value to rank your domain in search engines. Alexa rank can be improved by good content update and also by building backlinks from other relevant sources.