If I wanted to approach bloggers to see if I could pay them to write about my Website, what would be the best way to go about it?
You could use services like PayPerPost Direct, or Sponsored Reviews, or you could just leave a comment/email them about it. Most bloggers are reasonable people and willing to listen to what you have to say. Don't just spill the whole deal in the first contact though, just a short hey I like your blog, I was wondering if you would be willing to do a review of XXX. I would be willing to compensate you for your time. Sincerely, Me.
the mere fact that you're willing to pay them for doing something that they have been doing anyway will be enough attraction to your offer. be direct and sound sincere in your introduction. offer them an easy way to get the money, too, as a lot of people are more wary these days