Are you talking about a Chinese ID card?  Or for your home country?

If it is a Chinese ID card you are looking for, then it would be applied for in the same way as normal.  The child would need to be registered and then fill out the needed paperwork for the ID card.  Though, I don't know if they issue them for infants.

If you are talking about registering the child for your home country, you should contact your local consulate for details.  But, I would think that it required the natural mother and natural father to be married in order to gain foreign citizenship.  There may be ways around that, but it may require DNA tests to prove parentage.

"it"?

You refer to your child as "it"?

be aware that some countries will require you to register the child at the embassy/consulate within a certain time after being born. also if your child is registered with your country's embassy/consulate, then you will most likely also need to get you child a passport and have a visa made at your local PSB office.  

Children don't require or get ID cards, they go on the Hukou register. To do that you just need a Chinese birth certificate. When they are older (16 I think but not sure of exact age) they then use the Hukou to apply for an ID card.

Legally, if you apply for a 'foreign' (i.e non Chinese), passport the child will not be allowed an ID card when he / she is of age as the Chinese government don't allow dual nationality and technically they shouldn't be on the Hukou either as they are then deemed foreign nationals. (Although there are ways around that but they are illegal.)