The normal instruction, "Believe me!" is just an abbreviated form of "(You), believe me!" Inverting the phrase to "Believe you me!" is just a way of bringing the emphasis onto the word 'believe'.
It may sound rather archaic - though the earliest recorded use of the phrase dates back only to the 1920s - but it is not grammatically wrong.
Ted Hughes opens a poem with the words, "Terrifying are the intent, sleek thrushes on the lawn." Most of us, of course, would just say, "The intent, sleek thrushes on the lawn are terrifying." He, however, forces us to concentrate on the key word 'terrifying' by the inversion. Same with "Believe you me!"