日本人は他人にどう思われてるか、見られてるかを気にしすぎって英語でなんて言うの？

If we worry about what other people think of us, we make the conscious effort to act what we think is socially correct in public. We pay close attention to what we look like
and how we carry ourselves in public. This is called self conscious.

self-conscious
When you are nervous or uncomfortable because you are worried about what people think about you or your actions.
Ex.He looked uncomfortable, like a self-conscious adolescent

Japanese people are very conscious of other people's opinions
This is another good way of explaining how Japanese people always care about what other people think about them.

To be aware means that you would care about how others think about you.

These sentences above will help you to explain that said person believes Japanese people care a lot about what others think and their opinions of them,

Japanese people are overly conscious of how others perceive them.'

'overly conscious' is a way to say that someone thinks too much about something, or overly considers something.

'perceive' means view, look at, understand, regard, become aware of.

If someone is always worried about what other people think of them, then that translates into adopting behaviour that creates a favourable impression for others, or 'keeping up appearances.'

When a person cares too much about what others think, a term often used is self-conscious. A word to describe thinking or thought is perception. So, an individual may be self-conscious about their physical appearance (clothing, shoes, hair style), and have concerns for what someone else thinks or what someone else's perception is of them.