Use thee's and thou's in informal situations. If you're familiar with Romance languages such as Spanish, French and Italian, you probably know that these languages have two words each for "you." Thee's and thou's work in almost exactly the same way, as informal versions of "you." "You" was initially the formal version of the second person, while "thee" and "thou" were reserved for informal situations during the mid-15th to 17th centuries.
For example, in your screenplay, two characters who are close to one another may refer to each other with thee's and thou's. The same applies to an elder character speaking to a younger character, or a boss to a subordinate. When a character is speaking to someone with whom he or she is not close or when the character is speaking to someone of higher rank, you should use "you."
Avoid thee's and thou's when referring to more than one person. As with the second person in modern Romance languages, there is a second characteristic distinguishing thee's and thou's from you's besides informal versus formal usage. Thee's and thou's should only be used in the singular form -- i.e., when referring to one person. "You" is the plural form of the second person, so while it can be used to refer to one person in a formal situation, it's also employed to refer to more than one person, even in informal situations. Think of the modern equivalent of saying "you guys" or "you all"; "you" alone used to be all you needed to refer to more than one person.
Distinguish the usages of thee and thou properly. There are two versions of this antiquated informal, singular version of "you" because "thee" is the objective form and "thou" is the subjective form. (Even "you" used to have a second version, "ye," that was used in a similar fashion to "thou," but "ye" went out of fashion even before "thou" and "thee.") For a modern equivalent, consider that "she" is the subjective form of the feminine singular third person and "her" is the objective form.
A subjective pronoun stands as the subject of a sentence or clause. An objective form receives an action and cannot stand on its own. For example, you might write, "She gave a present to me" versus "I gave a present to her." When using thee's and thou's, write similarly. "Thou gavest [gave] a present to me" and "I gave a present to thee."