When should I say "thee"?

If I want to be posh, old school, when I'm writing, and decide to use "thee" then what is the correct technical usage for it? Does it simply replace "the" ?

To amplify what @Noah said:

Thee and thou are the second-person singular pronouns in English. They have fallen into almost complete disuse, as you are aware... but the funny thing is that most native speakers (of American English, anyway; I can't speak for British English speakers) have exactly the wrong impression of what they are for. The general impression seems to be that English has become less formal over time, and that we all speak familiarly to each other - but exactly the opposite is true!

In languages where the second-person singular is retained, it is only used between people who are extremely familiar with each other; if it's used in any other context, it's essentially an expression of dominance. (For example: a Spanish-speaking elementary-school teacher, in a tuteo country, would call his/her students "tú" - but woe betide the student who calls the teacher tú instead of Usted!)

Most (American) English speakers first meet "thee" and "thou" in the King James Version of the Bible, or in the works of Shakespeare. In the case of the Bible, we usually first hear the Ten Commandments ("Thou shalt...", "Thou shalt not...", "Thou shalt..."), and it gives us entirely the wrong impression. A much better starting point is the Lord's Prayer: "Our Father, Which art in Heaven, Hallowed be Thy name..." The clue here is that we are supposed to be talking to our actual father, and we call Him "Thee" because He's family.

So if you go to a Renaissance Pleasure Faire, or an event of the Society for Creative Anachronism, and you hear people "speaking forsoothly" (yes, that's actually what they call it!), prepare to be amused. In the actual Middle Ages, a commoner who addressed the Queen as "thee" would have been taught a rough lesson in short order.

In modern English usage, we don't use the second-person singular; what we use in its place is essentially the second-person plural - so it's not just Southerners who call each other "you-all"; we all do it! To see what I mean by this, consider the standard conjugation diagram, as seen in countless "501 xxxxish Verbs" books:

Modern English usage:

I    am   we   are   I    have  we   have    I    will  we   will   I   do    we   do
you  are  you  are   you  have  you  have    you  will  you  will   you do    you  do
he   is   they are   he   has   they have    he   will  they will   he  does  they do

Old usage:

I    am   we   are   I    have  we   have    I    will  we   will   I    do    we   do
thou art  you  are   thou hast  you  have    thou wilt  you  will   thou dost  you  do
he   is   they are   he   has   they have    he   will  they will   he   does  they do

In modern usage, we also use "you" as both the subject pronoun (who's doing it?) and the object pronoun (to whom is it being done?) Formerly, in the second-person singular, "thou" would be the subject pronoun, and "thee" would be the object pronoun:

Modern: I love you. Do you love me?
Old: I love thee. Dost thou love me?

Finally, thy and thine are the second-person singular equivalents of your and yours. To help you remember when to use which, try substituting "my" and "mine" instead.

Thy chair is on the right.
The chair on the right is thine.

Thine and mine are used in place of thy or my before a vowel sound, just like a and an.

This is a cap.
This is an apple.

This is my cap.
This is mine apple.

This is thy cap.
This is thine apple.

Here’s a brief history adapted from ‘The Cambridge Encyclopedia of the English Language’ by David Crystal:

Old English: thou singular, ye plural as subject. Thee and you as object. Middle English: ye and you used alongside thou and thee as polite singular forms. Early Modern English: Distinction between ye as subject and you as object disappeared, you being used almost universally. Ye restricted to archaic, religious or literary contexts by the end of the 16th century. Thou similarly restricted by 1700.

Ye is sometimes misunderstood as an archaic form of the, because of the resemblance of y to the Old English letter thorn, used to represent both the voiced and unvoiced 'th' sound.

Thee is a pronoun and is the archaic or dialect form of you not the. It's only used as second person singular compared to you which is used for both singular and plural.

We beseech thee O lord. We beseech you O lord.

Ref. Oxford dictionary.