You must be knowing, Wikipedia ranks high for its detailed and unique content supported with massive backlinks and other factors. If you need to try! Add quality content, Optimize your targeted keywords (Offsite and Onsite), fulfill your user needs and later try your luck.
I think you would beat Wikipedia only if you provided more details about the subject. Of course, this should be step 1. Step 2 is to get more links pointing to your page until you rank second (if you're not already there). After that, it should be a matter of time before Google recognizes your website (or page) and put it as #1 in the SERPS
And why would people want to beat Wikipedia in terms of ranking? Surely you wouldn't want to go that far just to beat Wiki for the keyword "cheese". LOL
Wikipedia has something called 'trust & Authority' which i'm sure your site will lack. It cannot be determined whether you will beat it for a particular keyword or not. But the way to go about is to not think about your competitors, just do SEO consistently and you will do well as far as you're doing it right. The caffeine update reduced the priority that Wikipedia received in the SERPs, so it also depends on future Google algorithmic updates.
I agree with the other posts on the how to do it, but why does it matter (other than pride?). People are either going to want wikipedia results or not. If they are not looking for results on wiki then a #2 ranking in SERP should be more or less as good as the #1 spot. IMO
A lot of people spend a whole lot of time on seo, me included. This is not the best way to the top. Create a great site that people love and they will join, participate and create content along with link to your site for you. Can you create a great site?