Okay so, as someone who has spent probably too much time in Blue Sentinels, lemme tell you. It kinda sucks.
It's a covenant with a really cool idea, but it doesn't really seem to work out as far as invading goes.
Okay so, if you're wearing the ring, you need to look up at your life bar. If the Blue Sentinel symbol is flashing, that means the game is searching in the background for people to invade. Theoretically, if a player in the Way of Blue covenant is invaded by another player, you will be summoned from wherever you are to help them, even across zones. In my experience, this is fairly rare.
You can also invade a "world of the guilty" with the blue eye orb. A guilty player is a player that has accrued "sin", by invading and killing other players or by killing NPCs. Even when the item isn't greyed out, you'll get "cannot find a world to invade" messages 90 percent of the time you use it. That said, when you use it make sure you're in a regular invasion/coop eligible area. It won't work in Majula for example, or other non combat areas.
Also, if you're in an area where you've beaten the boss, make sure to go to a bonfire and remove the effect of the human effigy; beating a boss now turns off invasions in the area for a half hour unless you cancel the effect.
The third thing you can do is take those tokens of fidelity you get from helping players beat bosses, and spend them to engage in duels with other Blue Sentinel players. There are statues next to the bonfire in the Cathedral of Blue.
