Theres a somewhat simpler way to do this using 2 standard SPDT automotive relays and some crimp-on female tab connectors. Using the left side as an example, tap a wire into the left front running light and run it to the 85 post on one of the relays then ground the 86 post. run the brake bright wire into the 87 post, the left rear blinker wire to the 87a post and wire the 30 post directly to the brake bright wire going into the brake light. Done, now repeat those steps on the right side, using the right sides blinker and running light wires of course.
When your turn the ignition on it toggles the relay to pass current from 87 to 30, which means that sides brake light works. When the blinker for that side is engaged the running light goes out and the relay defaults to it normal state, breaking the connection between 87 and 30 and connecting 87a and 30, letting the blinker control that sides bright brake instead, only the relay for the side thats blinking is affected so the other sides brakes works normally, one sides brakes are always toggled on no matter what.
This just uses standard automotive SPDT relays and a few crimp on female tab connectors to connect to the relays, it's actually pretty close to the same thing that Mitch has going on in his above post but done with commonly available relays. No soldering involved, just a few crimps. After working as a tech I really don't trust solder joints or wire twists much, seen alot of solder joints break and twists are, well, twists. A good crimper using non-insulated crimpon connectors and buttsplices then covered with some decent heat shrink tubing seems to do the trick for me. If you want to be fancy you can even buy the relay plug that connects to the relay and just wire all your connections into it, to each their own.
Anyways, I'll make a diagram and upload one in a few, it's really simple tho, the numbers I listed above are all stamped in the plastic on the relays next to all the posts.
Rez and Mitch's methods are great too, just all about whatever you have available to you to do it.

 

Just make the red connections.