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I replaced my tail light and rear turn signals with a combo LED strip and I wired in the appropriate resistors for the turn signals. All the wiring for this circuit is pretty simple. The turn signals work fine and the running light works but brake light only works when my headlight is on. Brake light does not work when headlight is off. Any ideas why??

Sounds like you may have your tail light power(headlight on and brake light works) feeding your brake light wiring somehow.

How could I have done that? Is that possible and it still work perfectly when the headlight is on?

Yes, it could certainly happen. If the brake switch grounds the circuit, then the power would be supplied to the circuit from the headlight circuit "upstream" of the brake light. If the brake switch provides the power, then the headlight circuit would be providing the power to the powered side of the brake switch.

If your headlight dims a lot when you hit the brakes, then the power to the brake light would probably be coming from the ground wire side of the headlight circuit, putting the brake light in series with the headlight. If it does not dim a lot when you hit the brake, then the power is coming from between the headlight switch and the headlight lamp, putting the brake light in parallel with the headlight.

I see what you're saying and that makes sense. I checked what you were talking about and I can't seem to figure it out. I have the brake light switch hooked up according to the wiring diagram and the brake light hooked up based on the same diagram. See wiring diagram link: http://bmwmotorcycletech.info/schematic1975,1976.htm