You've gotten a lot of good info above.
China indeed does not allow dual citizenships.
But you definitely- definitely- want your child to have citizenship outside China. If nothing else, it will make matters much easier when you decide to leave China.
And you WILL want to leave China. Sadly, as I have found out the hard way, having a child means that sooner or later, you must leave China.
Why?
1) Medical care here is disgraceful. Unreliable and inhumane.
2) The schools here are just awful. In addition to the rote-memorization-of-facts-you-can-never-question approach to education that we all come to know and love so much, schoolchildren here are subjected to horrendous loads of Party blather and anti-foreign propaganda. You want your kid coming home one day with one of those stupid red neckerchiefs around their neck? Of course not.
3) As a foreigner, it will be quite expensive and difficult for you to educate your child. Without a hukou, you'll have to pay through the nose to get them into school...there is no free education in China, and ESPECIALLY not for you and me and our kids. You will most likely be able to place your child into the so-called "noble schools" with the kids of the rich and well-connected. Not cheap...certainly not for you. WITH a hukou, your kid can go to the regular schools at a somewhat lower cost...but only in the city where the hukou is registered. A hukou very much ties you to your home town- indeed, this is its very purpose- and is not easy or cheap to change.
4) Your child will be subjected to relentless discrimination- by the homeys and especally by their schoolmates. My own child- who turns 4 next week as of this writing- is already beginning to get some of this in the schoolyard. You want to hear your child wishing they had a Chinese father like all the other kids?
I sure didn't.
5) Chinese society is rotten to the core. Corruption permeates everything here...there's no escaping it. Chinese grow up accepting this as simply the way things are, and an OK way for the world to be. I for one do NOT want my child growing up to believe these things.
An illustrative story: One day late in 2007 I was having a conversation (in Chinese) with my wife and our ayi. The housekeeper helpfully reminded me that I had better be sure to give my child's teacher a nice red envelope for Spring Festival with at least 800 RMB in it- else the teacher may not give the child enough to eat in the school cafeteria, or watch after her well enough to keep her safe and happy.
I simply don't want my child growing up to be part of a society where you have to bribe a nursery-school teacher to provide enough food to fill the belly of a 4-year-old. And I will wager that you don't either.
But, you say, aren't there pretty good international schools, and special Western-style hospitals with real doctors and stuff?
Sure there are...most larger cities in China offer these things. But you will find that even an upper-end English teacher's salary will not grant you access to these things. Go price a true international school some time and find out what I'm talking about. Even most of the managers, engineers, diplomats, etc. that send their kids to these schools don't pay these costs themselves...it comes as a bennie from their employers. The good hospitals, meanwhile, charge prices comparable to the costs of medical care in the USA...and as in the USA, no one can afford the care unless they have terribly expensive medical insurance. (NOTE- please don't confuse a Western-style hospital with the "VIP clinic" at the regular local hospitals. They're not the same thing. Most "VIP clinics" offer the same shoddy care that the homeys get, but it comes from a local doctor who speaks a little bad English.)
China can be a marvelous place to move to and live in as an adult. But you don't want to grow up here...and you don't want your child to grow up here.
Please get your child citizenship in your home country- while the getting's good.