Car running hot with A/C on

There's too many unknown variables for us to just guess at what the cause of your car troubles are, other than to suggest do some basic maintenance - check the serpentine belt, check coolant levels, check radiator cap/degas bottle cap, check if AC compressor is cycling on/off, and isn't seized. Thermostats can be easily installed backwards, hopefully that isn't the case here. If you have no coolant flow, however, that might be why. If you have a bad heater core, you might be telltale signs, such as antifreeze odor in passenger cabin, coolant leaking into cabin or on ground, misting from vents, and no heat in cabin area. It would not cause the engine to overheat, outside of leaking all the coolant out and thus causing engine to overheat. Check the simple things first, namely coolant level.