In my opinion the best way to raise awareness with the general public, not those who would seek out a website on their own, is to go where they are.
I would use the same technique Apple used to change minds when the iPhone was new. Come up with bite-sized ads, and play them like regular ads, many times, but instead of advertising for a product, they should educate.
The media is doing a terrible job at this. The same way Google tries to force ISPs to change with Google Fiber, we should try forcing the media to step up by putting them in a bad light, by focusing on the important problems they're ignoring.
Imagine 30 second ads that focus on non-partisan information, stats not emotions, factual information like current laws and how they're not followed, a focus on a more global picture, on something more long term, something that would make you think and be more aware.
There's a lot of misinformation out there. We know this from the numerous polls that show large number of people believing things that are factually and provably not true.
We need to correct that first.
EFF, ACLU, and other organizations like these are trying, but they're not reaching most people. We need to reach and educate the people who have been turned off by the current state of affairs.
We need Twitter-size ads that people would find interesting and that would wet their appetite to seek out more and get involved.