Hypercritically is the adverb: doing something in a hypercritical way. You look hypercritically at a piece of work when you hunt for tiny faults and give little credit for what is already good. Teachers, editors, and harsh relatives are the usual picture, but people also turn it on themselves. It tells you how the judging is done, not whether the person is fake. That 'fake' meaning belongs to a different adverb, hypocritically. It is a long, rather formal word. In speech, 'in a really picky way' will often do the same job. You will meet it more in writing than in chat.