Xiaoxing from Real Bout High School is a Chinese transfer student whose grasp of the Japanese language is somewhat limited. To be fair, she hasn't been in Japan for very long. However, even her thoughts are presented this way; She might just be a Third-Person Person. Also, as a subversion, she isn't treated as "inferior"; other characters don't even mention her language problems, and she's a valuable member of the group.
Shampoo from Ranma ½ is also Chinese, and speaks in a similar manner. Oddly enough her and the Jusenkyo Guide are the only Chinese characters who exclusively talk this way (Mousse is as Chinese as them and even talks in the Tohoku Regional Accent). In Shampoo's case, this is apparently blamed on learning a language by instructional tape rather than conversation; the official dub's adaptation of this quirk into broken English has the Unfortunate Implication of fans assuming she is much stupider than she is.
To be fair, she is The Ditz of a show where nearly every character is a first-class Jerk Ass.
Shampoo is actually a lot smarter then Fanon usually accepts her as being, she's just prone to Stupid Evil (as is everyone else in the series).
Ranma: Who'd want to date a stupid selfish cat girl like you?
This might just be Shampoo's Running Gag or something though. In the dub, her Mandarin is almost worst than her English! If she doesn't speak Mandarin or English well, what does she speak?!
Considering there are well over ten major "dialects" of Chinese that are generally less mutually intelligible than Swedish and Norwegian, or Swedish and German, she speaks her native language well enough. It ain't all Mandarin or Cantonese in China.
Ikuto/Keenan from Digimon Savers speaks Broken Japanese/English, due to having been raised by Digimon. However for some reason; the rest of the digimon speak perfect Japanese/English, including the two who were specifically his parental figures.
The dub takes this even further by giving Falcomon, Keenan's partner and adopted brother, an upper-class accent. And the two of them where apparantly raised together...
In Katekyo Hitman Reborn!, I-Pin's Japanese is atrocious; In the manga, this is conveyed by her speech being written sloppily to the point of being nearly unreadable. Admittedly, she's about the same age as Reborn and Lambo; She becomes fluent as she gets older.
I-pin's Japanese is not atrocious, in fact she does not even speak Japanese in the beginning. She speaks Chinese because she's from Hong Kong and later (10 years later to be exact) becomes fluent in Japanese.
The DelRey English translation renders this as Hulk Speak. As with Shampoo above, all Ku Fei actually does in Japanese is use the stock Chinese Verbal Tic — ending her sentences with de aru instead of desu.
Ling Yao of Fullmetal Alchemist hilariously invokes this when trying to weasel out of fixing the collateral damage brought upon the town of Rush Valley.
Edward Elric(points to Ling): This guy and his little entourage should be paying for everything!
Ling: So solly, I no understand much language of this country. Okay, bye-bye now!
Mr. Popo of Dragon Ball speaks like this in the original Japanese version of the series. It's dropped in the English version though in order to avoid Unfortunate Implications.
Simon from Durarara!! speaks very broken Japanese, and many people who run into him wonder if he's doing it on purpose to play up the "earnest foreigner" shtick. In his native Russian, however, he's very erudite and well-spoken.