How do I add a door to an already existing wall?
Add 2 inches to the door width to get your RO (rough opening) width. Add 2 inches to your height, to allow for a gap at the floor. (You probably won't need all of it, but sometimes level on the floor doesn't cooperate.)

If you absolutely know that the wall isn't load-bearing, then technically, you don't need a header. However, it's not hard to build a simple header, so I'd suggest you do it. 

If your wall is load-bearing, you'll want support while you've cut stuff away.

Other random tips:
Try really hard to get your hinge side studs perfect. If it's right, you can easily screw the hinge side of the jamb directly to the stud.
Cut an entire sheet of drywall away on each side, so you have lots of room to work on the framing. This also means that the new sheet you put in will have no seams around the door, which will help with cracking.