I'm going to answer your question but first I want to know why would you want to increase Alexa rank given it doesn't typically net any tangible value on its own merit?
I'll assume for a moment what you're really asking is “how do I increase traffic to my site” which there are many answers and right ways to go about doing that.
In summary and to answer both questions I would say build pages of quality content on your site, make sure your site is well structured using common web standards and basic SEO knowledge, then link to that content from other places around the web.
While your Alexa score may not improve as much as you might prefer, you'll notice an influx of high-quality traffic ready for any relevant CTA based on your industry/niche.