Full Valence Shell (Octet Rule) vs Formal Charge When does formal charge have more precedence over the octet rule / an atom having a full valence shell when drawing structures? 

When you draw $\ce{BCl3}$ for example, there are no double bonds - just three single bonds. Because of this $\ce{B}$ does not end up with 8 valence electrons. The explanation I have seen for this is that a double bond makes it so that the formal charge is undesirable. When drawing structures which has a higher precedence: formal charge or ensuring the atoms have full valence shells?
First, keep in mind that Lewis-structures and formal charges are huge simplifications.  In reality, at the quantum level, charges and bonds are spread out and it's somewhat unrealistic to say "Atom X has a charge of Q" or "Atoms X and Y have an Nth order bond between them".  There's merely a cloud of electron density, and it's somewhat arbitrary to say which parts of that cloud belong to a given atom or bond.

It's perfectly valid to draw a structure with a double bond.  And in reality, I'm sure there is hyperconjugation that leads to a small amount of double bond behavior.  However, such a resonance structure puts a negative charge on an electropositive element, and a positive charge on a very electronegative element, and thus such a structure has a smaller contribution to the overall structure.