1. Check the errors and recommendations in Google Webmaster Tools.This tool provides a lot of valuable information and tips about your website:
2. Monitor your backlinks with external tools
These tools provide the anchor text from your backlinks and the popularity (authority) of the websites linking to you. You should monitor this and strive to have a maximum of quality backlinks from high reputation websites/blogs with good keywords in their anchor text.
3. Analytics (Google Analytics or any other analytics software)
In your analytics, go to Traffic Sources > Search > Organic to monitor which traffic comes from search engines, excluding the visitors who found your site through paid advertising. Within this "Organic" segment, you will need to filter your keywords and exclude all the keywords related to your brand, because those visitors were already "looking" for you (with or without SEO).