A critique is a detailed evaluation of something. The formal way to request one is “give me your critique,” though people often say informally “critique this”—meaning “evaluate it thoroughly.” But “critique” as a verb is not synonymous with “criticize” and should not be routinely substituted for it. “Josh critiqued my backhand” means Josh evaluated your tennis technique but not necessarily that he found it lacking. “Josh criticized my backhand” means that he had a low opinion of it.
You can write criticism on a subject, but you don’t criticize on something, you just criticize it.