Your schematic should work for low PWM frequencies, your 'screaming', whatever that might be, is probably caused by a wiring error. Are you sure you connected the GND's of the 12V supply and the uC board together?
Other than that the circuit is suboptimal; it takes R3 a long time to charge the gate of M1 which increases dissipation. Replacing R1/R2/R3/Q1 with something like a TC4427 would be a better solution.
Regarding PWMming TEC's: you can do that as long as you keep the PWM frequency above 1kHz or so. If you PWM with a low frequency there is too much stress on the thermocouple joints due to them thermally expanding and contracting many times a second.