I want to know Greek root of exhilarating

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I don't understand what you mean by "just one word in proper English". If the root is Greek then it wouldn't be English, would it? Anyway, the etymology going back to Ancient Greek is detailed at https://en.wiktionary.org/wiki/exhilarate , for example.

It is not directly from Greek, but Latin.

1530s, from Latin exhilaratus "cheerful, merry," past participle of exhilarare "gladden, cheer," from ex "out, out of; thoroughly" (see ex-) + hilarare "make cheerful," from hilarus "cheerful"

The English word "hilarious" is from the same root.