How do I wire a relay board into existing series recepticles
The second picture shows the outlets wired in parallel, not series.

A relay is basically a electrically-controlled switch.  To turn off a normal outlet, all you have to do is interrupt the hot line, which black in your picture.  Disconnect the existing black wires and have a single black wire going to each socket.  Each of these black wires goes to one contact of the relay, and the master black connection to the opposite contacts of the relays.