First, I Want to comment on an answer I saw here ,
As there were some incorrect things in it.
1) Claim Your Site
This will not improve your rank in any way!
here is the proof from Alexa site :
Myth #3: Claiming my site affects its rank.
Claiming your site has no effect on rank and you don’t need to claim your site to get measured by Alexa. Claiming your site (which is a free service) and keeping your site description and contact information up-to-date ensures you won’t miss out on any potential business opportunities from the millions of people who visit Alexa every month.
2) Write A review About Actionable Analytics for the Web on your Site
I Don't know how this can help your Alexa rank, what it can be- once you could write
a review for a website in Alexa, this feature is no longer exist (at least from 2013)
3) Use Alexa Widget On your Blog
Maybe will help, not according to alexa.
Myth #4: Installing an Alexa widget will improve your rank.
No, the Alexa rank widgets allow you to promote your site’s rank to your site’s visitors. The widgets are not used to measure a site’s traffic.
4) Go for Alexa Pro
You don't need that for ranking, it can help you with overall traffic.
Myth #6: If I pay Alexa I can get a better rank.
Nope. Though it is true that Alexa’s Pro products can help you get more traffic to your website, which might in turn lead to a better rank. And, if you install the Alexa Certify Code on your website Alexa can directly measure your site’s traffic instead of estimating it. Direct measurement will give your site a more accurate rank, but more accurate does not necessarily mean a better rank.
What can Improve your Alexa rank ?
1) Install Alexa toolbar into your browser
This can improve, as Alexa panel consist from add-in toolbars (and more ). using the plugin in your browsers can get you counts. every visitor coming to your webpage with the toolbar installed, will also count.
2) Install Alexa Widget
Will a widget help you ? not according to Alexa.
Even so, I do recommend installing it , as there were bloggers showing improvement by using it and this a small thing you can do.
3) Write fresh high quality content
Maybe my best advice for you. content, content, content!
yes, this what can help you most to get more visitors, sales, better ranking and higher Alexa rank.Good content is what both users and search engines are looking for.
4) Write Original and engaging content
Engaging content is something that motivates people to do something with it. if you write that you're giving free ebook in your website and some one liked it, most changes he want to share it with a friend. that is engagement.
something that people need and seeking all the time.
5) Have relevant sites link to your website
Put it simple, backlinks. you want to have relevant, industry links pointing to you.
That's one sign telling search engines how good your site isthat's mean you have something.
6) Use driving traffic keywords
Shortly ,What is called on page seo. improve the keywords and optimize them on the page.
7) Buy Alexa Traffic
If you want to improve you rank shortly- buy traffic. but, as you stop it- expect to return soon to your original rank.
Have better SEO optimization
Work on all factors that matter to search engines. like: speed, mobile, keywords, etc...
What have you learned from my answer ?
Alexa want you to improve your rank naturally as possible. All the list points are things that can help you with this (except buy).
Last Question (maybe should be the first)
Why you want to improve your Alexa rank ?
You want that from the reason your website exist, maybe to sell something, to get people know you, to show expertise, etc...
So, please try to make your website better and helping people in what you can- this is the true way to improve your Alexa Ranking.