Writing interesting and keyword rick content is a strong point to anyone who has a good alexa rank. As long as your site is visible, that will bring in the better numbers for you.
So, what's now with high alexa rank? I don't think it will help a website rank well in search engines.
True, but from personal testing, after selling 2 websites in the same niche, but one with an Alexa around 100K and another one in 13 Million.. getting almost the same amount of visitors the first one sold almost immediatly, while the other one still isn't sold..
Basically Alexa rank is based on how much traffic you are getting and how many unique visitors visiting your websites. I think there is no shortcut to achieve quick alexa ranking
You forgot that they only count the traffic you get from people using their toolbar. So people not using their toolbar aren't counted. That's why it usefull to post stuff on webmaster forums, many webmasters have it installed so it'll get you a nice Alexa rank quite easily.
Alexa rank is useless, but I know alot of websites (10k Alexa and under), that won't affiliate with you unless you have an Alexa of relatively the same. Even if you have the same traffic, if not more.
For global rankings you just need any traffic. But if you want to rank high in your target country, maybe provide some content whose target audience come from that country.
Let the things on your SEO expert. Gain as much as traffic you can that is the only thing helps in Alexa ranking