How do I get past the overseers in High Oversser Campbell without being seen?

I’m having some real trouble with the first assassination. I’m where I saved the guy from the first overseer. I can’t get past the other 5 on the other side of the gate without confronting them. I tried to kill the one walking into the bunker, but then there was another one.

Any advice on how I can proceed?

By my estimation, you've just saved Overseer Martin, and are here, at this gate:

I like to Blink up to the air duct next to the gate. You have to jump and blink at the same time:

From there, you overlook an open area with three overseers. My preferred strategy is to Blink onto the tops of the street lights:

Blinking onto the next couple of lights gets you close enough to Blink to the stone ledge:

and from there across to the ledge of the building High Overseer Campbell is in:

You now overlook the main courtyard of the Overseer's building. Staying on this ledge will lead you to an open window where you can enter without confronting any enemies.

Ta-Da! You're in! Now to navigate the interior of the building.

Even if you haven't spent runes on upgrading Blink, I think this method will still work. Blink is probably the best skill Corvo has since it allows you to take so many new paths.