The best way to increase you keyword ranking is doing a good amount of relevant link building. You can try out guest postings, blog commenting and article marketing for the same. These all will contribute in ranking your website in the top searches of Google.
I also notice you want to search anonymously when you search for your own sites. Google might bump up your site if you click on it and/or are logged into your Google account.
I agree with Calvin. Get quality and relevant backlinks to your site. I would suggest reading and watching stuff about updated SEO. You can also get into seo trainings if you need more advanced skills or to really get you started on what works or what doesn't.