Using relay to control
You could do that, but instead of using the PS3's power to control the relay it's maybe more efficient to drive the LEDs directly with that. Standard red LEDs are about 2 V, then you can place 5 of them in series and add a current control resistor in series to fix the current at 20 mA. A relay would probably use double of that.

That being said, and though it's not likely that the PS3's power supply wouldn't have 20 mA to spare, you better use a transistor controlled by the PS3's power (doesn't have to be 12 V) to switch the externally powered LEDs. In that case you hardly load the PS3's power supply. 

 

The 5 V comes from the PS3, the 12 V is the external power supply. You'll have to tie the grounds of both power supplies together.