In general one doesn't expect to have nice necessary and sufficient conditions for checking positivity of a biquadratic form, although it may be possible to say something for small $n$. The sum-of-squares methods outlined in these course notes provide an efficient way of checking whether a given biquadratic form can be written as a sum of squares of bilinear forms. However, not all positive semidefinite biquadratic forms are sums of squares of bilinear forms as shown by Choi.
The sum-of-squares machinery gives a hierarchy of increasingly refined conditions which in the limit cover the interior of the cone of positive semidefinite biquadratic forms, so even if your form of interest is not a sum of squares of bilinear forms you may still be able to prove positivity by these methods, but you may not be able to do so efficiently.
Finally, note that the question of positivity of a biquadratic form is a first order statement over the reals and so is algorithmically decidable by quantifier elimination per Tarski, but the corresponding algorithms are exponential.