Presumably you'd like some answer less tautological that "a complete sequence is a frame when it satisfies the defining condition for frames," but then it isn't clear what your rules are.

Without loss of generality, you could take $\ell^{\Bbb N}$ for your $H$.  Then using your $f_n$'s as rows, a sequence takes the form of a matrix (with rows in $\ell^2$).  A priori, such a matrix defines an operator from $H$ to ${\Bbb C}^{\Bbb N}$.  Complete means kernel $\{0\}$; frame means bounded operator, with positive norm, to $\ell^2$.  So I read your question as asking for a characterization of boundedness and/or positive norm directly from the appearance of the matrix coefficients.  I don't believe that question admits any satisfactory general answer.