Firstly, ask yourself why you would want to? There is no benefit to it!.
However, if you insist on improving it, the only way to do so is to get more traffic to your site from those who have the Alexa tool bar installed in their internet browsers. There are a number of other claimed ways such as installing the Alexa widget, but none of those other ways actually work, as it all depends on traffic from those with the tool bar installed.
There are 3 way to get more traffic from those who have the Alexa tool bar installed:
1. Just improve the site, get more backlinks and do more basic SEO. As more traffic comes, a small percent of that increased traffic will have the Alexa tool bar installed and that will boost your Alexa ranking.
2. Install the tool bar in your browser and visit your site several times a day. Better yet, get a few friends to do the same thing. This will improve your Alexa ranking, but this also shows how easy it is to manipulate the ranking.
3. Use a paid Alexa improvement ranking service. This will usually work well in the short term, but you need to keep it up if you want to maintain the ranking.
Of course you don’t.
Too many webmasters obsess over there Alexa ranking. The Alexa ranking is a meaningless value that is of no use in helping a site rank in the search engines. The search engines simply do not use it.
Alexa works out a traffic ranking based on the number of visits to a site from those that have the Alexa tool bar installed in their browser. That is more than enough to see how it can be faked and how easy it is to manipulate. A lower Alexa score means that you are getting more traffic.
The only use of Alexa is if you want to impress a potential advertiser about the traffic and any purchaser of advertising who knows what they are doing will not be impressed with it as they know the problems with it!
If you want to use the Alexa ranking, then use it for its entertainment value and do not take it seriously. If you really do want to improve the Alexa ranking, there are ways to do it.