I am optimizing my site for 3 keywords right now, one is almost complete will start on the other 2 today. I prefer to do it one at a time so just waiting for next backlink update from google to wait and see the results.
When starting a backlinking campaign you should never focus on a few keywords. Focus on ALL that apply including the name of your business. Search engines are getting smart. They have been smart but now they are implementing their ideas. The Google patent has just been released and it states clearly that generating links for the sole purpose of gaining SERPs can result in your website being banned. What you have to do is use common sense. If you only target a few phrases, search engine spiders will recognize the intent meaning it's obvious you were asking the webmasters how to link to you.
No, the more content you add and the more links are coming in, the wider a variety of related keywords you can rank for. There will be a core term that your rankings are built around and the more power your site gains, the broader the circle is going to be.