This version of Koszul duality (as well as many others) can be deduced from the Barr-Beck-Lurie theorem (cf. Lurie's book Higher Algebra). You can consider the functor
from k-mod to A-mod of tensoring by V and ask for it to have a left or right adjoint, giving rise to a comonad or monad on k-mod (ie the coalgebra or algebra form). These are two finiteness conditions on V (k-dualizability and A-dualizability - ie dualizability as a vector space or compactness as A-module - though maybe I'm mixing up the order). If you ask for both simultaneously then you guarantee that the functor from A-mod to (co)modules over the (co)monad is both limit and colimit preserving -- this is overkill but an easy way to ensure the hypotheses of the Barr-Beck-Lurie theorem are satisfied. You now get an equivalence between your category of (co)modules and a COMPLETION of A-mod --- i.e., the part of A-mod that your particular chosen module V sees (the category it generates). (This is how you satisfy the other requirement of the theorem, i.e., that the corresponding functor is conservative).

The classical version of this is A=S, symmetric algebra in one variable, ie A-mod = quasicoherent sheaves on the line, and V is the augmentation module (skyscraper at the origin). We then have a descent theorem, saying that the completion of A at the augmentation is derived equivalent to sheaves on a point with descent data for inclusion of point to line -- which is exactly modules for the Koszul dual exterior coalgebra (or dually exterior algebra).