I can say "An American" or "A Frenchman", however, can I say "A Chinese" like that? Does it sound weird?
Yes, you can say "a Chinese" but yes, it sounds at least a little weird to many people most of the time. This is discussed in Why can we say 'an American' but not 'a British'? I'd recommend using the adjective "Chinese" instead it in situations where it is easy to do so, but the noun "Chinese" can be used when necessary. You can see this use listed in a number of dictionaries, such as Merriam-Webster and the American Heritage Dictionary which have as one definition of Chinese "a native or inhabitant of China".
Here is an example I found from the book Sweet Bamboo: A Memoir of a Chinese American Family by Louise Leung Larson (accessed via Google Books):
Being a reporter had not occurred to me; I did not think a Chinese, and especially a woman (women reporters were rare in those days), had the remotest chance.
Alternatives that can be used in certain circumstances:
As a predicate, the adjective can be used instead: "He/she is a Chinese" can be replaced with "He/she is Chinese."
When speaking of Chinese people in general, "the Chinese" is a fairly common phrase, using what seems to be the same grammatical structure as "the French", "the rich", "the poor", "the blind". It takes plural agreement.
Of course, "Chinese person/Chinese people", or if you are talking about a specific person of known gender, possibly "Chinese man" or "Chinese woman", are always possible alternatives.
Another note: the plural form of the noun, "Chineses," was used at one point in time (you can see some results on Google Books from the 1600s-1800s), but it sounds completely wrong to my modern ears. I can't think of any circumstance where I would use it, or advise using it.
There are other nouns that have been used in the past to refer to Chinese people that now sound dated: I would strongly advise against using them as they may cause offense. They are Chinaman (constructed similiarly, though not quite the same way as Frenchman) and Chinee (which stems from a reinterpretation of "the Chinese" as a plural).
Pretty much the same goes for all other nationality words ending in "-ese".
Other somewhat similar topics, although certainly not identical: