I have a domain name and website. The name can be spelled differently,
but the different spelling is already registered.

It has been registered for several years, but there has never been a
live site at the domain.

I have offered to purchase the domain, but they don't want to sell it.

When you type it in, what comes up is a search page. When you type the
name into google without the www. & .com, it brings up lots of things,
but not my site.

I know sometimes webmasters include common misspellings so that when
people type the misspelling into a search engine, their site comes up.

My question is: would this work for me? If so, how do I do it? (i.e.
in the meta tag? In the body?)

There is not only one little answer to that. The process is called SEO.

If you like to learn all the facts read 
http://www.seochat.com/c/b/Search-Engine-Optimization-Help/

In the meantime, by all means, add the misspelling to your META TAG.

Anyone can do this for almost any domain. You optimize one page just
for the term. I would suggest you you check our the seo articles at:
http://www.websitetrafficbuilders.com/search-engine-optimization-seo.htm
and they have a free trial software that can assist you and you may
never have to purchase the full version.