The correct English is "Do I look Chinese?" It is incorrect to add the "like".

长得像 is another thing. 我长得像中国人 means you have a chinese appearance: black hair, black eyes, yellow skin etc. 我像中国人 or 我看起来像中国人 means you do something that makes people feel you are a chinese OR you have a chinese appearance. For example, you are a foreigner but you always eat chinese food, you do everything like chinese people do, now we can say: 你像一个中国人.

我长的像中国人吗？ I think this maybe more local.

我像一个中国人吗？ --- OK 我看起来像一个中国人吗？ --- OK