For one of my supplements, I currently am using the lines:
"Each time I visited him [my grandfather], we would have an intimate conversation full of the broken english I remember so well:
“Anon, I told you how I leave China?”
“Yes you have, Anon.”
“Oh? Well then I tell you now!”"
English is not his first language, he came to the US from China, and thus talks similarly to how I wrote it above. Since his english isn’t grammatically correct, should I paraphrase instead of quoting? Or find another way to write it? Are those quotes okay, or would they be too stereotypically Asian and come across to some as racist (that is how my grandfather talks)?
You are not writing a novel (long dialog is more suitable for that purpose), so which way is most convenient to convey the message about your grandfather to the admission office, please use it. Beware that the supplemental essay is still about you and not your grandfather, so anything about your grandfather needs to tie back to what inspire you from him for what your activities or characters you want to put into supplemental essay.
This looks fine to me and actually quite charming. It’s perfectly acceptable to quote incorrect English—it portrays how characters really talk and makes them seem more real. The rest of your narrative will make it clear that you know proper grammar.