Remove the quick couplings from the heater core pipes at the firewall, and drain as much coolant out as you can, then to avoid a mess inside the car, blow compressed air through one side and push the rest out... make the mess under the hood where it is easily rinsed away. Do not have your mouth open and staring at the pipes when you push compressed air through them, or you will get blasted.
Then, loosen the screw down between those two pipes. Loosen, do not remove it. This allows the pipes inside the car to wiggle around a bit so you can get the core out.
You'll still dribble a little coolant inside the car, have a towel laid down. But it won't be much.
There are o-rings on the connection tubes at the core inside, replace those if they look damaged.