What are the best strategies to expand your brand using Google Plus?

Here are a few tips to grow your brand on Google+

1. Set up rel author to all your blog content - this way you can link your G+ account to external content - https://support.google.com/webmasters/answer/2539557?hl=en
2. Join in relevant communities on Google+ - https://plus.google.com/communities 
Join in discussions on the communities and make your self known.
3. Follow relevant people in your industry if you see a chance to help them do it, it can mean re shares of content or any thing.
4. Start your own community on Google+ and share content via it, I run a community with 7k members.
5. Try and get included in relevant niche specific circles on Google+ people make them and then share them via the network you can gain 100s of followers via this.

Any more specific questions please ask me, I have been using Google+ since the day it was released ;)

In addition to Jame's answer, Since your Google Plus account will show up on Google searches, so make sure you keep posting things and up to date!

I would also suggest using Google Hangouts - it creates a video for your youtube channel as well lifting your SEO ratings.