Is "believe you me" proper English?
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.