Your circuit is mostly correct. The common would be the 12V keyed voltage source (do not connect it to the battery directly unless you like paying for new ones weekly). The normally closed would be the DRL, and the normally opened the flasher circuit. You'd need one that can deal with the low currents of an LED strip.
The control switch and "in" is where your circuit isn't accurate. Most common relays do not have a separate input from the positive and negative poles of the coil. The switch would be in line with the relay coil power. Plus an inline Fuse. That's the only change.
simulate this circuit – Schematic created using CircuitLab
Typical automotive relay:
Alternatively, there are flasher modules that include the relay, so no separate relay is needed.