Translation:I came from China.
Yes. There are two uses of から. This one following a noun is under the group "case particle" 格助詞（かくじょし）. A case particle is used to mark the noun before the particle with a role in the sentence. e.g. が is used to mark a noun as a subject. から is used to mark a noun as the origin of an action.
Just to avoid creating additional confusion, "I was born in China." would be 「私は中国で生まれました。」 The verb 生まれる (うまれる) means "to be born".
In contrast, the noun 出身 (しゅっしん) means "origin" and it can be used to refer to your home town, home country, school/college, etc.
So when you say 「私は中国出身です」, you are literally saying "I am of Chinese origin."
In the example「中国から来ました」 the verb 来る ("to come") and the particle から ("from") are used to indicate that you just came from the location. "I come from China." This could mean that you are native to China, but it also might mean that you are a non-native who recently traveled from that country. Also, you obiously would not use this phrase if you were currently IN the country of China. That would sound weird.
The very basic answer is です is only really used for certain sentence structures. The overly simplistic answer is です translates as "to be" or "am" or "is" or things along those lines, so if you've seen the sentence 中国出身です, it literally translates more like "I am a person from China" (Not quite but again, being overly simplistic)
For pretty much every other verb (In this case, "to come"), they end with ます in the present tense, or ました in the past tense.
"I came from China," might be the literal translation of this sentence, but it seems a strange way to translate it, given the sorts of contexts that I am imagining. What is the most common way in which the Japanese sentence is used? Is it mostly used for saying where you are from? If so, then, "I came from China," would not be the right way to translate it.
If someone asks you where you are from, and you wanted to reply in good English, you might say, "I'm from China." You would never say, "I came from China."