here are my suggestion:
1 Produce engaging, original conten
2 Get more related sites to link to your website
3 Find keywords that are driving traffic to your competitors
4 Understand your competitors’ marketing strategy and learn what works
5 Follow SEO best practices on your website
Alexa Rank is more of your visitors are getting alerted of the Alexa Ranking. The alexa improved due to your traffic increase, not the opposite way. And alexa ranking can be boost by using tools, it is not accurate enough. Alexa Ranking provides you an immediate view of you and your competitors’ performance comparison.
A regular update schedule will keep readers coming back to your site again and again, and will show the search engines that your site is active. If you're writing a blog, try to update weekly at least.