What is wrong with my L7805CV + BUF 654 NPN transistor circuit
You said:


  I have check the regulator and it works fine when it's not connected to the transistor, but I get strange low readings when I hook it up to the transistor. Taking out the 5V wire and connecting it again blinks the LED strip shortly.


This is a sign that you are overloading your regulator. 7805 regulators have built-in thermal protection, which means that as they are getting hotter, their voltage sags and eventually refuses to operate. When you remove the power supply, the regulator cools just enough so that when you plug it in again, it turns on momentarily.

More troubleshooting questions:
Is your circuit hooked up right? You currently have the 7805 turning on the transistor, which then allows current to pass from the 12-volts into the LED strip and to ground. What's the point of having the 7805 regulator then?

Also, the reason why the 7805 is getting too hot!
The 7805 is connected to the base of the transistor, while the emitter is connected to ground. the base-emitter junction of the transistor acts like a diode.

If you want to do this you must have a current limiting resistor. Without it, your circuit essentially shorts the output of the 7805 regulator to the ground.

I would simply recommend removing the transistor altogether. The 7805 is rated for 1 amp, already more than what the 12-volt input can handle, which is 500mA, so you really don't need the transformer to boost the output. That should do the trick.

Personally, If I were to do this particular project, I would use an alternative voltage-drop solution. I would connect multiple diodes in series. Each diode drops the voltage by 0.6 volts. This prevents unwanted "waste" heat caused by the regulator, and would be much simpler to implement.