Backlinks from high PR websites is only way to increase Alexa.

Increase your website traffic and organic search

Best Idea to improve Alexa rank is by increasing the website traffic. More the unique traffic of website help you to Improve your Alexa rank. Other Alexa rank easily improved by enabling the Alexa toolbar on website.

In SEO field "Alexa" provides complete information about your website.
It provides the local & global ranking of a particular website and also analyze and show the percentage of visitors of your site.

Few tips for improving Alexa rank of your site-

Add niche of your blog.
Frequently updating a blog.
Giving backlink to Alexa.
Comment on blogging niche blogs.

Alexa rank of your website is based on your website traffic and Bounce rate for the last three months, Once your website traffic is good and have good bounce rate, your Website Alexa rank will be decreased.

I just don't know what is the point of increasing your Alexa ranking how is it going to help you. Although I check my website Alexa ranking after every two days but I don't think so it is going to help you in any way. It is just to check how much visitors is your website generating.