The Nightmare in North Point DLC allows you to get three extra achievements (two are unlocked by getting through the missions, the other is for completing everything), the Zodiac Tournament DLC doesn't have any achievement.
I bought Sleeping dogs when it went on sale a month ago, without any DLC. I LOVE the game and decided to buy the following DLC today. I would recommend playing the game without ANY DLC and then buying them later.
I think it would ruin the game, having over powered weapons/cars/clothing. That's just my opinion though, I would recommend buying the game and then buy the DLC as a gift, so you can unlock them later
I recommend the same, you won't get a feeling of progression if you activate the DLCs immediately. You'll get more out of Sleeping Dogs if you buy them as a gift and unpack them after you complete the story.
Buying Zodiac Tornament and Nightmare in North Point will not ruin the single player game, since Nightmare is a standalong and I dont think the zodiac gives you any big advantage. Buy all of the other interesting DLC as a gift, or not at all. If you hate the game, you will not want the other DLC
Zodiac Tournament - integrated with the base game, unlike Nightmare in North Point. It's fun if you like the martial arts part in Sleeping Dogs, should last you 30 minutes or so but I believe it's worth $1.25
Nightmare in North Point - lasts two boring hours, repetitive as hell, save file not compatible with the base game. Only worth it if you completed everything in Sleeping Dogs and are craving for more.