How Do I Change The Heating Core In My 2003 Dodge Ram 1500?

You will have to remove the evaporator to replace the heater core which means you have to remove the whole dash. very long and hard job to do . you will have to remove freon to change it.

On a 2001 Dodge Ram Pick-up with a 6 cylinder engine

First you have to have the freon removed,go to a auto repair shop to have this done. And ask how much was removed so when you have it charged back up it should only cost what you need plus labor.
Second,Buy a good repair manual and fallow every step by step. It's a pain in the a$$ because the manual will have you jump from chapter to chapter on everything you remove.I replaced mine in 8 hours and that was taking my time.If I had to do it again I could easily do it in 5-6 hour.
Tools you will need. Everything on the dash uses only two sizes torx drivers,make sure you buy a set with long shafts you will need it on the steering wheel. You also need the little tool that separates the freon lines,it looks like a small donut made out of plastic.It's just a few dollars.
Get yourself one of those plastic knife looking things that glass repair guys use on windshield trim,that works great for removing a lot of dash parts that require popping off.
  Good luck.