Depends on what your blog is about and what goals you have for the blog.
In general, make sure your content is the highest quality you can make it. Be interesting, entertaining, provide useful information, make people laugh, make people think -- whatever it is that's the purpose of your blog, be the best you can be at it.
Then, network with other like-minded bloggers. If yours is a food blog, get to know other foodies. If yours is a humor blog, get to know other online comedians. If yours is a corporate blog, network with other related businesses that also have blogs. Share information, get to know at a personal level the individual (or individuals) who write(s) their blog.
Be generous with your links. When you find an interesting article on someone else's blog, write a blog post with your own original commentary (not just a summary of what they posted, but what you think about it yourself) along with a link to the original article.
Use social media to promote your blog. Make sure you tweet out whenever you publish a new post, and that you include it on your Facebook and Google+ pages (and LinkedIn, if it's a business-related blog). If you've got a visually oriented blog, post pictures to Pinterest. And so forth. Whatever platforms are appropriate for promoting your blog, use them. And don't forget to include links on the blog itself to make it easier for others to follow you and share your content on their own social streams.