Suppose $H$ is a Hilbert space with orthonormal basis $\{e_i\}_{i}\in \mathbb N$ and $X=H^*\otimes^\pi H$ is the projective tensor product. We have a natural isometry $$J:X\to X^{**}=B(H)^*$$ given by $J_{\sum k_1x_i\otimes y_i}(T)=\sum k_i\left<T(y_i),x_i\right>=\sum k_iT(x_i\otimes y_i)$ for all $T\in (H^*\otimes^\pi H)^*=B(H)$
I am looking for linear functionals which are not in $J_X$. One such example is $\psi:B(H)\to \mathbb C$ defined as $$\psi(T)=\lim_{n, U}\left<Te_n,e_n\right>$$ where $U$ is any is non-principal ultrafilter on $\mathbb N$ , which is a well defined bounded linear functional which do not lie in the image of map $J$. Is it a complete charecterization of functional which do lie in $J_X$? If not what are the other ways we can get hold of such functionals?
"Not being something" is often something that is not pretty to characterize.
Here you can get an example by mixing a normal part and a non-normal part. Take one of your $\psi$, and define $$ \varphi(T)=\psi(T)+\langle Te_1,e_1\rangle. $$ This is still not normal, but now it is not zero on the rank-one projection $e_1\otimes e_1$, so it cannot come from an ultrafilter.
There might be other ways to construct non-normal functionals, but I don't know about them. You are asking to fully characterize the dual of the Calkin Algebra; I don't know if that's feasible and/or known.