If you haven't read the first part, do it now! It should take at least two weeks to get your kitty (and you!) comfortable enough to move on to part two.
By now, your cat (hopefully) should be wearing his or her harness like it's nothing. In addition, you want your cat to come running to you whenever you hold up his or her harness. This means Kitty is now associating HARNESS with HAPPINESS. Good job.
Let's work on leash training. While there are some very special cats who will happily follow you while on a leash, others won't. A cat's natural inclination is to walk and/or dash in seven different directions at once. Even worse, if they feel any pullback from their leash, they will either crouch down in misery or flail around like the ground has suddenly turned into ice, fire, or water!
Step 6: Put your cat's harness on and attach the leash. Don't go outside yet. The two of you are going to practice the art of walking straight. Hold the leash loosely in one hand and plenty of yummy treats in the other. Take one step forward and hold out a treat to get Kitty to walk forward too. Keep doing this one step at a time until you run out of treats. Eventually, progress to two or three or even more steps.
Step 7: When your cat veers to the side, don't jerk back on the leash. Just stay still and let the leash grow taut on its own. Coax Kitty back to your side with a treat without pulling on the leash. You want Kitty to learn that the leash will tighten only if he or she wanders away from you and it will slacken as long as he or she remains by your side. You want to keep working on this until Kitty gets used to being attached to a big, awkward human and will (reluctantly) follow you around the house while on leash.
Step 8: Take a deep breath. It's time to go outside. If you have a backyard, start there. Pick a time when it's quiet. Evenings are great because cats feel safer when it's darker but it's not so dark that you can't see what's happening. Get Kitty all harnessed up like usual then walk him to the door. That first step out into the big world can be really scary so go slow. You might have to spend this training session just staring through an open door, and that's perfectly fine.
Step 9: You're outside. Okay. The outside world is filled with distractions, most of which you can't even see or hear but your cat can. In the beginning, let Kitty explore to his or her heart's content but always with you holding onto the leash. If he or she starts wandering into an area you'd like to avoid, just let the leash grow taut like in Step 7. Slacken the leash as soon as Kitty switches direction. If you've got a stubborn one, pick him or her up and place back down in a more appropriate location.
Step 10: Once the novelty starts to subside and Kitty starts paying attention to you again, repeat Steps 6 and 7 while outside. Eventually, you'll be leash walking your cat around the neighborhood like a pro!
Remember: Not all cats will be able to make it to Step 10, and that's OK. Your cat will tell you when he or she has had enough. While leash training a cat requires a lot of patience, the process should ultimately be rewarding for the both of you. It's great if you can get to the point where Kitty runs to the door whenever you say, "Let's go for a walk!" but it's also great if Kitty prefers to stay inside the house. The reward is getting to spend quality time with each other. You're strengthening your bond and learning something new together. I guarantee your cat will be happier because of that—even if Kitty never sets a single paw outside.