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 $\ell^{2}$ 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.