Hi Mehmet.
Firstly, the BDX53 is not a MOSFET, it is a bipolar Darlington transistor.
The voltage you need to fully turn it on is Vbe(sat) which the data sheet says is 2.5V at 12mA, assuming 3A collector current.
The transistor has a pair of resistors across the B-E pins which draw some of that current. So if you are producing PWM at 5V amplitude (5V supply to the PIC) the resistor in series with the base can be no bigger than (5V - 2.5V)/0.012 = 208 Ohms. If you are using a 3.3V supply it has to be no bigger than 91 Ohms. Your 10,000 Ohms is way too high! Be aware that dropping the resistor will present a risk to the PIC if the transistor C-B junction breaks down, it would be wise to add a Zener diode across the PIC output to help protect it.
The maximum voltage on the collector depends on a number of things, mostly to do with the transformer. It will certainly be higher than the supply voltage by a factor of 2 or more.
Brian.