Ask HN: What is the best way to advertise my first Chrome extension?

A post here called "Show HN: Crowd Reviewed Websites (Chrome Extension)" would be a good place to start. You can set up facebook and google+ pages where you give information on your extension, and have these point to a blog where you deliver progress updates on developments. Once you've got the pages up, tweet links, get friends/existing users to like and share the links so that more people become aware. If it's something people are interested in the news will spread. If it's not spreading fast enough through that, be a bit more creative. Contact people who produce good content (e.g. popular bloggers/YouTubers, other indie developers, tech journalists) asking for their help. Some may like your product so will publicise / link to / review / for free, others you may need to offer incentives to (what that is depends on who they are and what you can offer them).

It's also a good idea to create a brand and have your extension as part of that brand. Publicise the brand more than the extension. The reason for this is you may have other projects in future - you don't want to start from scratch each time. Once you have the brand, you can talk about other products within that brand's context, and people who've subscribed to your brand because of your previous work will then be informed about your new stuff.