Solution 1:

The phrase "Believe you me" copies the archaic word order one finds in Early Modern English for a marked imperative. Typical examples are from King James version of the Bible (both testaments).

See e.g. Book of Matthew 14:16

They need not depart; give ye them to eat.

and in a few common phrases such as "mind you" (but with a slight nuance) for example

Not that I would have accepted her offer, mind you!

This is also very common in Shakespeare.

Romeo and Juliet, Act 3, Scene 4, Lord Capulet speaking:

And bid her, mark you me, on Wednesday next--

[...]

Well get you gone: o' Thursday be it, then.

In the interrogative voice, it takes an accusing turn.

As you like it, Act 5, Scene 2, Phebe speaking:

If this be so, why blame you me to love you?

and my favorite, Hamlet, Act 4, Scene 2, Rosencrantz speaking:

Take you me for a sponge, my lord?

It seems "Believe you me" is a relatively recent recreation of this syntax if one believes my copy of the OED.

Solution 2:

Wiktionary has references from the 1840s and 1870s so this is old enough to register as a well-established idiom. I had some trouble finding other uses of "[verb] you me" until a blog pointed out the King James Bible:

For thus saith the Lord unto the house of Israel, Seek ye me, and ye shall live

Using Verb-Subject-Object order is "an archaic form used for imperatives." The blog also found references from earlier than those on Wiktionary and goes on to hypothesize (emphasize added):

The interesting observation is, at first glance, that the majority of these early examples come from Irish publications [...]. A hypothesis: perhaps it isn't a relic of the archaic English VSO construct, but arrived by a different route. Irish Gaelic is a VSO language; maybe its sentence pattern influenced Irish English? A quick Google suggests "Believe you me" might correspond to the Irish phrase "Creid uaim é!" = "Take it from me!" (literally, "believe from-me it").

Solution 3:

It's real English.

It's based on archaic English grammar, e.g., phrases like "Hear ye me" in the King James Bible.