ContiKnitter's site is good. I have looked for sites to explain duplicate stitch before and they don't go very far in their explanations. Regarding the picture in the site, once you get the stitch to the point in the picture, put the needle back down through the center of the stitch you came up in to start with, and go directly over to the center of the next stitch to the left, if you can. You don't always get to work that way. BTW Once you get started it is almost all done from the front.
Did you do the heart intarsia, and have only the letters to do?
I usually begin working right to left because I'm right handed and it is more natural to work that way.
I have a Leisure Arts booklet on duplicate stitch from 1991 and it says it is best to work left to right or right to left. I take that to mean to work across one way and then back the other. It would be a real pain in the neck to have to start a new thread with each pass so that you could go left to right all the time (although it would probably be ideal). It's kind of like working left handed to go from left to right, but it can be done.
The booklet gives the advice that it is best to work horizontally, but if you have to work vertically (such as when you change rows [they say]) to work from top to bottom. They claim that working bottom to top can distort the knit stitches. The last thing I duplicate stitched was for a paid job of putting a name on a Christmas stocking for someone and I wanted to do it the best I could so I started at the top of the chart and worked back and forth and down. It turned out fine.
Your design is tricky with all those long vertical lines and not much in between to join them. It is fine to carry the yarn a little ways from one stitch to another, but I wouldn't do that too far. Maybe it would be good to do each letter separately working right to left and left to right from top to bottom, carrying the yarn across the places it needs to be for doing the indivdual letters. If this is on a scarf or something where the back is going to show you might want to do the vertical lines vertically. The book says top down is best, but I don't know if you have to go by the book. It may look fine if you go up. You may have to experiment on a swatch and see if you see any real difference.
If you are doing the heart duplicate stitch, I guess from what the Leisure Arts booklet says, I would start at the top and do it back and forth working downward. Then do the letters. And I don't see a real easy way to do them "by the book". You'll just have to try and see what works best for you, and consider how much "stuff" you want to have showing on the back. I'm lazy about having a bunch of different ends to deal with and tend to bend the "rules" much of the time.
I hope this helps a little, and maybe someone with vast experience with duplicate stitch will have a really good answer. Good luck.