It's a good idea to monitor hashtags that are close to your blog topic, either on Twitter or Facebook or both the best. It's good way to find people that are interested in the same topic as you.
Join a few Facebook groups that are similar to your blog topic, and join in the conversation in there. Some groups allow just plain promotion, some ask for a proper conversation with other group members. Whatever it is, don't be too aggressive in promoting.
Over time, and if you have a really good post that you're very proud of, invest in some ads. They aren't too expensive but could help in growing your audience.
Don't forget to add sharing options on your website, so people can share it fast and easy.
At the end, as many others said, it's all about making the content that is interesting, educational and worth sharing.