Summarize:

Prop open the door as you change the lock. Turn the screw under the door latch counterclockwise with a flathead screwdriver and remove it. Put the screw somewhere safe because you'll have to screw it back in later. Insert the key into the keyhole and turn it slightly so that it's in the open position. This will hide the latch in the internal components of the lock and allow you to slide it out. If you don't turn the key then you won't be able to pull out the lock. Keep the key turned while you push on the lock from the other side of the door. If done correctly, the interchangeable lock cylinder should come out. If the cylinder feels stuck, try turning the key a little more or less. Adjust the key until you can push the latch out. Turn the key counterclockwise a quarter of the way open so that the latch goes inside of the lock cylinder. This will allow you to easily slide it back into the cylinder hole. Thread the screw into the hole with your fingers, then use a flathead screwdriver to tighten it. Continue turning it until it runs flush with the side of your door. Screwing in the flat screw will hold the lock cylinder in place. Keep the door open as you test it. If it doesn't work, try dissembling and reinstalling the lock cylinder or call a locksmith to see if they can replace it for you.
Remove the flat screw under the latch. Turn the key a quarter of the way. Pull on the key while pushing on the other side of the lock. Turn the key and slide the new cylinder into the hole. Screw the flat screw back in. Test your key to make sure that it unlocks your door.