From Adams, we know that the algebra of (unstable, degree-zero) cohomology operations $K^0(BU)$ can be written as formal infinite linear combinations of canonical generators 

$$\mu_n := \sum_{i=0}^{n} (-1)^{n-k}\binom{n}{k} \psi^k$$

However, from the collapse of the Atiyah-Hirzebruch spectral sequence for BU, we also know that $K^0(BU) \cong \mathbb{Z}[[c_1^K,c_2^K,\ldots ]]$ where $c_i^K$ are the Conner-Floyd Chern classes (where I'm renormalizing them to degree zero by an appropriate power of $t\in \pi_2KU$).

Thus, it should be possible to write the Chern classes in terms of the Adams operations. How can I find these expressions?

Doing the reverse is not so bad, using Hirzebruch's theory of genera: I get that $\psi^k$ is $(1+c_1^K+c_2^K+\ldots)^k$. But unfortunately I'm completing lacking in the power series wizardry that would allow me to invert this.